Here’s an animation I made to celebrate.

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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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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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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

and from triangle *EDC* we get

And since

we can conclude that

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

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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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.

]]>This is a tricky question. So tricky, in fact, that it tripped up those responsible for creating this exam.

Dilation is a similarity mapping (assuming, as we do, that the scale factor is non-zero), and translation is a congruence mapping. Thus, any composition of the two will be a similarity mapping, but not necessarily a congruence mapping. So in the above question, statement II will always be true, and statements I and IV are not always true.

Statement III requires closer attention. Under most circumstances, translations and dilations map lines to parallel lines, and so the same would be true of their compositions. However, if the center of dilation lies on a given line, or the translation is parallel to the given line, then that line will be mapped onto itself under the transformation.

This means that the answer to this test question hinges on the question, “Is a line parallel to itself?”

If the answer is yes, then statement III will always be true, and so (3)* II and III* will be the correct answer. If the answer is no, then statement III won’t always be true. and so (1) *II only* will be the correct answer.

So which is the correct answer? Well, that’s tricky, too. The answer key provided by New York state originally gave (3) as the correct answer. But several days later, the NYS Department of Education issued a memo instructing graders to accept both (1) and (3) as correct. Apparently, the state isn’t prepared to take a stance on this issue.

Their final decision is amusing, as these two answer choices are mutually exclusive: either statement III is always true or it isn’t always true. It can’t be both. Those responsible for this exam are trying to get away with quietly asserting that (*P* and *not P)* can be true!

Oddly enough, this wasn’t the only place on this very exam where this issue arose. Here’s question 6:Notice that this question directly acknowledges that the location of the center of dilation impacts whether or not a line is mapped to a parallel line. It’s not entirely correct (a center’s location on the line, not the segment, is what matters) but it demonstrates some of the knowledge that was lacking in question 14. How, then, did the problem with question 14 slip through?

As is typical, the state provided a meaningless and generic explanation for the error: this problem was a result of *discrepancies in wording*. But there are no discrepancies in wording here. This is simply a careless error, one that should have been caught early in the test production process, and one that would have been caught if production of these exams were taken more seriously.

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Consider this multiple choice item from the June, 2017 Common Core Geometry exam.

Instead of testing a student’s understanding of triangle congruence, this question exposes a serious lack of mathematical understanding among the exam creators.

A superficial reading of the problem suggests that (3) is the correct answer. In (1), the two triangles share only three pairs of congruent angles; in (2), two sides and a non-included angle are congruent in each triangle; and in (4), the triangles share only one pair of congruent sides and one pair of congruent angles. None of these scenarios (*AAA*, *SSA*, *SA) *seems sufficient to guarantee that the triangles are congruent. And in (3), the triangles have one pair of congruent sides and two pairs of congruent angles; this (*ASA** or SAA*) is sufficient to conclude the triangles are congruent, so (3) is apparently the correct answer.

But closer inspection shows that, in fact, (1) and (2) are correct as well.

Consider choice (1). While it’s not exactly clear what it means to map angle *A* onto angle *D*, it must require that point *A* gets mapped to point *D*. Similarly, point *C* must be mapped to point *F*. If a rigid motion maps *A *to *D* and *C* to *F*, then segment *AC *must be congruent to segment *DF*. We now have one pair of congruent sides and three pairs of congruent angles: the triangles are congruent (by *ASA* or *SAA*), and choice (1) is a correct answer.

In (2), we are given that segment *AC* is mapped onto segment *DF. *This means that point *A* gets mapped to point *D* and point *C* gets mapped to point *F*. And since segment *BC *is mapped onto segment *EF,* we know that *B* is mapped onto *E*. Therefore, the vertices of triangle *ABC* are mapped via rigid motion onto the vertices of triangle *DEF. *This is sufficient to conclude that the triangles are congruent, and choice (2) is also a correct answer. (It’s also worth noting that, since the triangles are given as acute, *SSA *is actually sufficient to guarantee that the triangles are congruent. This mathematical error turned up in a separate question on this exam.)

As it stands, the only option that is not a correct answer to this question is (4).

Within a few days, the NYS Education Department issued a directive to count all answers to this question as correct. As is typical, no admission of an error was made: the problem was blamed on *discrepancies in wording*. Of course, there are no *discrepancies in wording* here: this problem as written, reviewed, edited, and ultimately published is simply mathematically incorrect. Its existence demonstrates a fundamental misunderstanding of the underlying concepts.

This isn’t the first time an erroneous question has made it onto one of these high-stakes Regents exams. In fact, there were at least three mathematically invalid questions on this exam alone! Over the past five years I’ve documented many others, and each time it happens, it raises serious questions: Questions about the validity of these exams, how they are experienced by students, how they are scored, and the lack of accountability for those in charge.

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Founded in 2011, the mission of 100kin10 is to recruit, train, and retain 100,000 excellent STEM teachers in 10 years. To that end, they lead a coalition of nearly 300 public and private-sector organizations committed to supporting STEM education in the US.

Over the past two years, 100kin10 has been working to identify the key systemic problems in STEM education. Today’s STEM Grand Challenges launch maps out the interconnected landscape of those problems and issues a series of challenges that address the problems identified as most pivotal among the network.

At the event, I’ll be speaking about my personal and professional experiences with some of the grand challenges, and I’m looking forward to hearing from a variety of different perspectives on how we can best improve STEM education.

You can learn more about 100kin10 here.

]]>Celebrate Permutation Day by mixing things up! Try doing things in a different order today. Just remember, for some operations, order definitely matters!

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The theme of the conference is *Big Ideas In and Out of the Classroom*. I’ll be delivering the opening keynote, *Connecting Big Ideas In and Out of the Classroom*, and will also be running a related workshop later in the conference. I’m honored to be providing the opening remarks for this event, which is undoubtedly the first of many.

The Summer Think has been organized and orchestrated by teachers from Math for America’s various fellowship programs, and features two dozen sessions proposed and facilitated by teachers. In addition to the underlying support offered through their fellowship programs, MfA has provided organizational support for the conference.

Teacher-led, teacher-driven professional development lies at the heart of Math for America’s programs, and the fact that over 50 teachers are participating in a conference less than two weeks after the end of the school year speaks to the impact it has on MfA’s teachers.

You can learn more about the conference, including a list of sessions and presenters, here.

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