Archive of posts filed under the Uncategorized category.

## Regents Recap — June 2016: Still Not a Trig Function

I don’t know exactly why, but fake graphs on Regents exams really offend me.  Take a look at this “sine” curve from the June, 2016 Algebra 2 Trig exam.

Looking at this graph makes me uneasy.  It’s just so … pointy.  Here’s an actual sine graph, courtesy of Desmos.

Now this fake sine curve isn’t nearly as bad as these two half-ellipses put together, but I just don’t understand why we can’t have nice graphs on these exams.  It only took me a few minutes to put this together in Desmos.  Let’s invest a little time in mathematical fidelity.

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## Regents Recap — June, 2016: Reused Questions

When I looked at performance data from the June 2016 Common Core Geometry Regents exam, I noticed that students did exceptionally well on one of the final multiple choice questions.  It didn’t take long to figure out why:  it was virtually identical to a question asked on the August 2015 exam.

Just a few words changed here and there.  All the specifics of the problem, and all the answer choices, are exactly the same.

The most basic quality control system conceivable should prevent questions from being copied from last year’s exam.  It’s hard to understand how something like this could happen on a high stakes exam that affects tens of thousands of students and teachers.

Issues like this, which call into question the validity of these exams, are what led me to originally start asking the question, “Are these tests any good?”

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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.

This 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.

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.

## Regents Recap — June 2014: When Good Math Becomes Bad Tests

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

It is a true geometric wonder that a triangle’s medians always intersect at a single point.  It is a remarkable and beautiful result, and the fact that the point of intersection is the centroid of the triangle makes it even more compelling.

This result should absolutely be a part of the standard Geometry curriculum.  It important and beautiful mathematics, it extends a fundamental notion of mathematics (symmetry) in new ways, and it is readily accessible through folding, balancing, compass construction, and coordinate geometry.

But here’s what happens when high-stakes testing meets meaningful mathematics.

This wonderful result has been reduced to an easy-to-test trick:  the centroid divides a median in a 2:1 ratio.

It’s not hard to see how such a fact can quickly become an instructional focus when it comes to centroids:  if that’s how it’s going to be tested, that’s how it’s going to be taught.  Of course, teachers should do more than just teach to a test, but there’s a lot riding on test results these days, and it’s hard to blame teachers for focusing on test scores when politicians, policy makers, and administrators tell them their jobs depend on it.

This is just one example of many, from one test and one state.  This is an inseparable component of standardized testing, and it can be found in all content areas and at all levels.  And for those who argue that the solution is simply to make better tests, keep this in mind:  New York has been math Regents exams for over eighty years.  Why haven’t we produced those better tests yet?

## Calculus Gave Me a Speeding Ticket

Years ago, one sunny Sunday afternoon, I was driving home from visiting friends at college and received a speeding ticket.  I didn’t realize it at the time, but calculus played an important role in my citation.

You see, this was no ordinary speeding ticket, the kind where a police officer paces the offender or uses radar to measure a vehicle’s speed.  My speed was calculated from an airplane high above the road.  And the Mean Value Theorem clinched the case.

Aerial speed enforcement works like this:  large marks painted on the road divide the highway into quarter-mile intervals.  A pilot flying overhead uses a stopwatch to time a suspected speeder from one mark to the next.  Say the pilot records a time of 12 seconds; a simple calculation converts one quarter mile per 12 seconds into 75 miles per hour; this information, the average speed on this interval, is radioed to the police on the ground who then stop and ticket the driver.

What I didn’t realize at the time was how crucial calculus is in all of this.

A fundamental theorem of calculus, the Mean Value Theorem (MVT), relates the average rate of change of a function with the instantaneous rate of change of the function.  Suppose we have some function of time, $f(t)$, and suppose that we know the value of this function at two times, say $f(t_1)$ and $f(t_2)$.  The average rate of change of $f(t)$ between $t_1$ and $t_2$ is

$f_{avg} = \frac{f(t_2) - f(t_1)}{t_2 - t_1}$

The MVT tells us that, as long as $f(t)$ is a differentiable function, then at some time between $t_1$ and $t_2$, say at t = c, the instantaneous rate of change of $f(t)$ must have been equal to the average rate of change of $f(t)$ from $t_1$ and $t_2$.  That is,

$f'(c) = \frac{f(t_2) - f(t_1)}{t_2 - t_1}$

where $f'(x)$ is the derivative of $f(x)$, the instantaneous rate of change of $f(x)$.

What does this have to do with my speeding ticket?  Well, as I’m moving along the highway in my car, the pilot records two values of my position function, $x(t)$, at two different times, $t_1$ and $t_2$.  The pilot then computes my average speed

$x_{avg} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}$

Here’s where calculus comes in.  The Mean Value Theorem says that, at some point between those two times my instantaneous speed must have been equal to my average speed.  If my average speed was above the legal limit, then at some time between $t_1$ and $t_2$, my instantaneous speed must have been above the limit, and at that moment, I was guilty of speeding.

I wonder if it would have helped to argue that my position function wasn’t differentiable!