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8/15/17 — Happy Pythagorean Triple Day!

Today we celebrate a rare Pythagorean Triple day!  And especially rare, as 8-15-17 is a primitive Pythagorean triple.  We won’t see many more of those!

Here’s an animation I made to celebrate.


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Regents Recap — June, 2017: The Underlying Problem with the New York State Regents Exams

I’ve been writing critically about the New York State Regents exams in mathematics for many years.  Underlying all the erroneous and poorly worded questions, problematic scoring guidelines, and inconsistent grading policies, is a simple fact:  the process of designing, writing, editing, and administering these high-stakes exams is deeply flawed.  There is a lack of expertise, supervision, and ultimately, accountability in this process.  The June, 2017 Geometry exam provides a comprehensive example of these criticisms.

The New York State Education Department has now admitted that at least three mathematically erroneous questions appeared on the June, 2017 Geometry exam.  It’s bad enough for a single erroneous question to make it onto a high-stakes exam taken by 100,000 students.  The presence of three mathematical errors on a single test points to a serious problem in oversight.

Two of these errors were acknowledged by the NYSED a few days after the exam was given.  The third took a little longer.

Ben Catalfo, a high school student in Long Island, noticed the error.  He brought it to the attention of a math professor at SUNY Stonybrook, who verified the error and contacted the state.  (You can see my explanation of the error here.)  Apparently the NYSED admitted they had noticed this third error, but they refused to do anything about it.

It wasn’t until Catalfo’s Change.org campaign received national attention that the NYSED felt compelled to publicly respond.  On July 20, ABC News ran a story about Catalfo and his petition.  In the article, a spokesperson for the NYSED tried to explain why, even though Catalfo’s point was indisputably valid, they would not be re-scoring the exam nor issuing any correction:

[Mr. Catalfo] used mathematical concepts that are typically taught in more advanced high school or college courses. As you can see in the problem below, students weren’t asked to prove the theorem; rather they were asked which of the choices below did not provide enough information to solve the theorem based on the concepts included in geometry, specifically cluster G.SRT.B, which they learn over the course of the year in that class.”

There is a lot to dislike here.  First, Catalfo used the Law of Sines in his solution: far from being “advanced”, the Law of Sines is actually an optional topic in NY’s high school geometry course.  Presumably, someone representing the NYSED would know that.

Second, the spokesperson suggests that the correct answer to this test question depends primarily on what was supposed to be taught in class, rather than on what is mathematically correct.  In short, if students weren’t supposed to learn that something is true, then it’s ok for the test to pretend that it’s false.  This is absurd.

Finally, notice how the NYSED’s spokesperson subtly tries to lay the blame for this error on teachers:

“For all of the questions on this exam, the department administered a process that included NYS geometry teachers writing and reviewing the questions.”

Don’t blame us, suggests the NYSED:  it was the teachers who wrote and reviewed the questions!

The extent to which teachers are involved in this process is unclear to me.  But the ultimate responsibility for producing valid, coherent, and correct assessments lies solely with the NYSED.  When drafting any substantial collaborative document, errors are to be expected.  Those who supervise this process and administer these exams must anticipate and address such errors.  When they don’t, they are the ones who should be held accountable.

Shortly after making national news, the NYSED finally gave in.  In a memo distributed on July 25, over a month after the exam had been administered, school officials were instructed to re-score the exam, awarding full credit to all students regardless of their answer.

And yet the NYSED still refused to accept responsibility for the error.  The official memo read

“As a result of a discrepancy in the wording of Question 24, this question does not have one clear and correct answer. “

More familiar nonsense.  There is no “discrepancy in wording” here, nor here, nor here, nor here.  This question was simply erroneous.  It was an error that should have been caught in a review process, and it was an error that should have been addressed and corrected when it was first brought to the attention of those in charge.

From start to finish, we see problems plaguing this process.  Mathematically erroneous questions regularly make it onto these high stakes exams, indicating a lack of supervision and failure in management of the test creation process.  When errors occur, the state is often reluctant to address the situation.  And when forced to acknowledge errors, the state blames imaginary discrepancies in wording, typos, and teachers, instead of accepting responsibility for the tests they’ve mandated and created.

There are good things about New York’s process.  Teachers are involved.  The tests and all related materials are made entirely public after administration.  These things are important.  But the state must devote the leadership, resources, and support necessary for creating and administering valid exams, and they must accept responsibility, and accountability, for the final product.  It’s what New York’s students, teachers, and schools deserve.

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Regents Recap — June, 2017: Consistency and Precision

Two prominent themes of my critical review of the New York State Regents exams in mathematics are consistency and precision in language.  Here’s a pair of problems from the June 2017 Geometry exam that illustrates both issues.

First, the phrasing of the question “What is the number of degrees in the measure of angle ABC?” is awkward and somewhat unnatural.  Second, if we are going to ask for “the number of degrees” in the measure of an angle, then the answer should be a number.  The answer choices here are not numbers: they are degree measurements.

Why not simply ask for the measure of the angle, as was done in question 10 on the exact same exam?

While the issue in question 4 is minor, we know that imprecise use of language is deeply connected to student misconceptions in mathematics.  And we know that an important part of our job as teachers is getting students to use technical language correctly.  Our exams should model the mathematical clarity and precision that we expect of students in our classes.  Far too often, the New York State Regents exams don’t meet that standard.

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Regents Recap — June, 2017: When Side-Side-Angle is Enough

Here is yet another mathematically erroneous question from New York’s June 2017 Geometry Regents exam.

At first this question seems straightforward.  There are several ways to determine if two triangles are similar, and the answer choices cover three of the basics: in (1) segment AB is parallel to segment ED, so congruent alternate interior angles can be used to show that the triangles are similar by Angle-Angle (AA); in (3) Side-Angle-Side (SAS) similarity can be used; and in (4), Side-Side-Side (SSS) similarity applies since all three pairs of sides are in proportion.

Presumably (2) is the answer choice that does not guarantee the triangles will be be similar, and according to the official scoring guide provided by the state (2) is the correct answer.  But as it turns out, (2) is also sufficient to guarantee that the triangles are similar.  This means that this question has no correct answer.

In (2), we have two pairs of sides in proportion and one pair of congruent angles (the vertical angles ECD and ACB).  This is the Side-Side-Angle (SSA) scenario, and because this set of information does not determine a unique triangle, SSA alone is not sufficient to establish that a pair of triangles are similar (or congruent).

But there is additional information to work with in this question.  The lengths of the sides of the triangles guarantee that angles B and D are both acute.  This is because there can be at most one non-acute angle in any triangle, which is necessarily the triangle’s largest angle, and the largest angle in a triangle must be opposite the triangle’s longest side.  Since angles B and D are not opposite their respective triangle’s longest side, they must be acute angles.  And it turns out that this additional piece of information allows us to conclude that the triangles are similar.

Here’s why.  Suppose you know the lengths of segments XY and YZ and the measure of an acute angle Z.  Depending on the length of XY, there are 0, 1, or 2 possible triangles XYZ.  Here’s a geometric representation of all the possibilities:

This explains why SSA fails to uniquely determine a triangle:  there may exist two different triangles consistent with the given information.

But if two triangles XYZ are possible, one of the triangles will have an obtuse angle at X and the other will have an acute angle at X.   This means that if we happen to know that angle X is acute, then only one triangle XYZ is possible, and so this set of information (SSA and the nature of the angles opposite the given sides) uniquely identifies a triangle and can be used to establish similarity (or congruence) among a pair of triangles.  Thus, the information in (2) is sufficient to conclude the triangle are similar, and so there is no correct answer to the above exam question.

Alternately, a more algebraic argument uses the Law of Sines.  From triangle ABC we get

\frac{sinB}{7.2} = \frac{sinACB}{8.1}

sinB = \frac{8}{9} sinACB

and from triangle EDC we get

\frac{sinD}{2.4} = \frac{sinECD}{2.7}

sinD = \frac{8}{9} sinECD

And since

\angle{ACB} \cong \angle{ECD}

we can conclude that

sinB = sinD

Generally speaking we can’t conclude that the measure of angle B is equal to the measure of angle D:  two angles with the same sine could be supplements or differ by a full revolution.  But since we know both angles are acute, we can conclude that

m\angle{B} = m\angle{D}

Thus, the triangles are similar by AA.  (This argument also shows that SSA together with knowledge of the nature of the angles is a congruence theorem.)

So, this high-stakes exam question has no correct answer.  And despite the Change.org petition started by a 16-year old student that made national news, the New York State Education Department refuses to issue a correction.  In fact, they refuse to acknowledge the indisputable fact that this question has no correct answer, perhaps because they don’t want to admit that a third question on this exam (see question 14 and question 22) has been determined to be mathematically erroneous.

UPDATE:  All the media attention apparently convinced the NYSED to award full credit to all test takers for this erroneous question.  Due to the discrepancy in wording, of course (link).

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07/21/2017 — Happy Derangement Day!

Today we celebrate a Derangement Day!  Usually I call a day like today a Permutation Day because the digits of the day and month can be rearranged to form the year, but there’s something extra special about today’s date:

The numbers of the month and day are a derangement of the year:  that is, they are a permutation of the digits of the year in which no digit remains in its original place!

Derangements pop up in some interesting places, and are connected to many rich mathematical ideas.   The question “How many derangements of n objects are there?” is a fun and classic application of the principle of inclusion-exclusion.  Derangements also figure in to some calculations of e and rook polynomials.

So enjoy Derangement Day!  Today, it’s ok to be totally out of order.