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Regents Recap — January 2015: Admitting Mistakes

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

This is question 27 from the Geometry exam.

January 2015 GEO 27This question has two correct answers:  it is possible to map AB onto A’B’ using a glide reflection or a rotation.  The original answer key indicated that (4) glide reflection was the correct answer.  After the exam was administered, the state Department of Education issued a correction and told scorers to award full credit for both (2) and (4).

Mistakes happen, even on important exams that many people work hard to produce.  But when mistakes are made, those responsible should accept responsibility, not equivocate.

Here’s the official correction from the state.

January 2015 GEO 27 -- Correction

It’s hard to accept that the issue here was a lack of specificity in the wording of the question.  The issue is that someone wrote a question without fully thinking through the mathematics, and then those tasked with checking the problem also failed to fully think through the mathematics.  This isn’t a failure in communication; this is a failure in management and oversight.

And it has happened before.  This example is particularly troubling, in which those responsible for producing these exams try to pretend that an egregious mathematical error is really just a lack of agreement about notation.  Sometimes errors are just erased from the record with little or no explanation, and then, of course, there are the many mistakes that are never even acknowledged.

Mistakes are bound to happen.  But by pretending that substantial errors are just misunderstandings, differences of opinion, or typos, the credibility of those responsible for these high-stakes exams suffers even further damage.

An Ancient Tiling Problem

This menacing creature is Glyptotherium texanum.  Well, was Glypotherium texanum.

Glyptotherium Texanum

The armor caught my attention.  It’s made of small plates that create an interesting tiling of a curved surface.

Glyptotherium Texanum up close

Notice how the tiles slowly change from quadrilaterals to pentagons to hexagons, and back again!  I wonder what factors determine the shape of each tile as they grow.

Regents Recap — January 2015: It’s True Because It’s True

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

This is question 25 from the Common Core Algebra exam.

January 2015 CC A 25

I’ve already complained about the contrived, artificial contexts for these questions (why not just ask “Is the sum of these two numbers rational or irrational?”), so I’ll ignore that for now.  What’s worth discussing here is the following sample student response provided by the state.

January 2015 CC A 25 -- Sample Response

So, why is the sum of a rational number and an irrational number irrational?  Because the sum of a rational number and an irrational number is always irrational.  This circular argument is offered as an example of a complete and correct response.

I’m not sure there’s a way to rewrite this question so that it admits a sensible answer.  That’s probably a good indication that it shouldn’t be on a high-stakes test.

As I’ve argued time and again, questions on these exams should stand as examples of proper mathematics.  But questions like this actually encourage bad habits in students, and teachers too, who are being told that this constitutes an appropriate response to this question.  This is yet another example of the danger of simply tacking on “Justify your reasoning” to a high-stakes exam question.

Regents Recap — January 2015: Not Even Pseudo-Context

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

This is question 8 from the Integrated Algebra exam.

January 2015 IA 8Four students are playing a math game at home.  One of the math game questions asked them to write an algebraic equation.

The context of this question is utterly absurd  The question might as well have been

“Four students are taking a math test.  One of the questions asked them to write an algebraic equation.  Which student answered the test question correctly?”

Why not just ask “Which of the following is an example of an algebraic equation?”.  Maybe there are people who believe that framing questions as games, or humanizing them, will engage test-takers more, but it’s hard to believe that contrivances like this do anything but further separate students from the concepts they purport to assess.

Math teachers are familiar with the notion of pseudo-contextbut I’m not sure what I would call this.  Meta-pseudo-context?  Pseudo-meta-context?  Pseudo-pseudo-context?  Ridiculous, at the very least.




Field Goals, the Super Bowl, and Mathematical Models

There has been a tremendous increase in the use of mathematical analysis to make policy, inform organizational decisions, and explain news and events.  I generally think this is a positive thing:  I understand that math gives us a powerful set of tools for understanding and processing the world.  But I also understand the limitations inherent in mathematical modeling.

All mathematical models rely on assumptions that limit their impact.  Mathematicians are typically aware of the assumptions about objects, relationships, and contexts that their models make.  Politicians, journalists, and others who invoke mathematics to make a point seem less aware.  This often leads to bold, unjustified claims based on what “the math” has told them.

An inconsequential but illustrative example of this occurred during Super Bowl 49.

At the start of the second half with the game tied at 14, Seattle drove into scoring position, and faced a 4th-and-1 at New England’s 8 yard line.  Seattle basically had two options:  kick the field goal (a high-percentage play for 3 points), or try to make a 1st-down, and ultimately a touchdown (a lower-percentage play for 7 points).  Seattle opted for the field goal and went ahead 17-14.

Lots of people on Twitter second-guessed the decision, including The Upshot’s David Leohnardt.

Here, David Leonhardt is applying a simple expected value argument.  “Going for it” on 4th-and-1 at the opponent’s 8-yard-line likely produces more total points in the long run than kicking field goals, which suggests that Seattle should have gone for it.  It’s not a bad argument; in fact, I used a similar analysis on the NFL’s new overtime rule.

But in order to apply this argument, it’s important to understand what assumptions this model makes.  For example, this model assumes that the amount of time remaining in the game is irrelevant.  Of course, it’s not:  it’s easy to construct a situation in which “time remaining” is the determining factor in the kicking a field goal (say, the game is tied, and only seconds remain).

This model also assumes that all points are of equal worth.  But they aren’t.  Depending on the game situation, the extra four points a touchdown gives you may be irrelevant, or of significantly less value than the three points the field goal gives you (imagine a team up by six points late in the game).

There are lots of factors this analysis does not consider.  This doesn’t mean that the expected value argument is invalid.  It just means that, like all mathematical models, what it says depends on the assumptions it makes.  And the more we use mathematical models to drive our decisions, the more important it is to be clear about the assumptions that are made and the consequences they entail.