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Regents Recap — June, 2017: Trouble with Dilations (and Logic)

The emphasis on transformations in Common Core Geometry has proven to be a challenge for the creators of the New York State Regents.  Here’s the latest example.

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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Regents Recap — June, 2017: More Mathematical Misunderstanding

Far too often questions on New York State math exams demonstrate a disturbing lack of content knowledge.

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 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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MfA Summer Think 2017

Graphic for WebsiteI’m excited to be participating in the inaugural Summer Think, a teacher-led summer conference hosted by Math for America.

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.

Making Math with Scratch — Scratch Ed

Scratch Ed, an organization at the Harvard Graduate School of Education that supports teaching and learning with Scratch, recently profiled some of my work teaching mathematics using Scratch.

The article, Making Math with Scratch, highlights a Math for America workshop I ran for teachers that centered on approaching mathematical concepts through the lens of coding and computer science.  Several projects I use in my classroom are featured, and I also discuss why I like teaching with Scratch and how it’s become a valuable part of my approach to teaching math.

The purpose of the workshop and Patrick’s classroom activities are to demonstrate the power of bringing mathematics and computer science together. “Ultimately the goal is to show how math and computer science are great partners in problem solving. And Scratch provides a terrific platform for that.” 

I’m excited to share the work I’ve been doing with math and Scratch over the past few years–including talks and workshops at conferences like Scratch@MIT, SIAM ED, and the upcoming NCTM Annual meeting–and I really appreciate this nice profile from Scratch Ed.

You can read the full article, Making Math with Scratchat the Scratch Ed website.

PAEMST Applicant Webinar

On Wednesday, February 15th, I’ll be participating in a webinar hosted by the National Science Foundation for teachers who are applying for the Presidential Award for Excellence in Mathematics and Science Teaching (PAEMST).

The purpose of the webinar is to help interested teachers navigate the application process, which involves a lot of planning, recording, reflecting, and writing.  As a PAEMST awardee, I’m looking forward to sharing my experiences applying for, and receiving, the award.

If you are applying for the PAEMST, or considering it, you can register for the free webinar by clicking here and searching for “Applicant Webinar”.  I will be participating in the webinar on 2/15 at 2:00 pm, but NSF is running webinars throughout the nomination period, so there are many dates and times to choose from.

And if you know a deserving teacher, it’s not too late to nominate them for the Presidential Award!  You have until April 1st, 2017.

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