Presenting at MOVES Conference

moves logoI am very excited to be a part of the inaugural MOVES conference at the Museum of Mathematics in New York City!

The focus of the conference is the Mathematics of Various Entertaining Subjects, and it features an amazing lineup.  Erik Demaine, Dave Richeson, and Henry Segerman are among invited speakers, and Tim Chartier and Colm Mulcahy will be part of special evening of mathematical entertainment!

I will be running a Family Track activity at the Museum on Monday afternoon.  This workshop, Sphere Dressing, is inspired by the activity I submitted for the 2012 Rosenthal Prize.

The conference runs August 4-6.  You can find out more information here, and see the entire conference program here.

Regents Recap — June 2013: Encouraging Bad Habits

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

Some of the worst exam questions aren’t merely erroneous, but actually encourage students to exercise bad mathematical habits.  Consider these questions from the June 2013 Algebra 2 / Trig exam.

2013 June A2T 4

Notice that the problem doesn’t specify what kind of series this is:  the student is expected to assume that the series is geometric.  This is a terrible habit to encourage, and I wrote about this the last time it happened on a Regents exam.  I guess no one is listening.

2013 June A2T 3

In order to determine whether a relation is one-to-one or onto, it is necessary to know the relation’s domain and range.  Here, the student is expected to assume that the oval on the left represents the domain of the relation and the oval on the right represents the range.  Perhaps these assumptions are reasonable given the nature of the diagrams, but this just seems sloppy to me.  I wouldn’t accept imprecise formulations of functions and relations like this from my students; I would demand they be more explicit.  (It’s also worth noting that the relation in (2) is one-to-one and onto its image in the right-hand oval.)

Here’s another example of sloppiness in question construction.

2013 June A2T 19

Is it supposed to be obvious that the i here is the imaginary unit?  The letter i could be just a variable, like, say, the m that also appears in the question.  The available answers support the assumption that i^2 = \sqrt{-1}, but why are we forcing students to play test-detective?

The Regents exams also continue with their long-standing tradition of presenting unscaled graphs, another bad mathematical habit to encourage.

I believe these tests should stand as exemplars of proper mathematics.  Maybe I’m alone in thinking this, but it seems to me that repeated exposure to these sloppy exam questions might actually interfere with a student’s ability to truly understand the underlying mathematics.

Regents Recap — June 2013: More Trouble with Functions

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

Functions seem to be an especially challenging topic for the writers of the New York State math Regents exams.  After this debacle with functions and their inverses, we might expect closer attention to detail when it comes to functions and their domains and ranges.  We don’t seem to be getting it.

Consider this question from the June 2013 Algebra 2 / Trig exam.

2013 June A2T 33

According to the rubric, the correct answer is -900a^2.  This indicates that the test-makers either (a) don’t understand the concept of domain or (b) they have decided to start working in the world of complex-valued functions without telling the rest of us.

Let f(x) = ax \sqrt{1-x} and h(x) = x^2, and note that g(x) = h(f(x)).  In order to evaluate g(10), we first have to evaluate f(10).  But f(10) = 10a\sqrt{1-10} = 10a\sqrt{-9}, which isn’t a real number.  Thus f(10) is undefined; in other words, 10 is not in the domain of f(x).

But if 10 is not in the domain of f(x), it can’t be in the domain of g(x) = h(f(x)) either.  Therefore, g(10) is undefined; it is not -900a^2, as indicated in the rubric.

Of course, if we are working in the world of complex numbers, \sqrt{-9} = 3i.  But we never talk about complex-valued functions in Algebra 2 / Trig.  When we talk about functions like g(x), we are always talking about real-valued functions.  And just because the process of squaring later on down the line eliminates the imaginary part, that doesn’t fix the inherent domain problem.  After all, what is the domain of f(x) = ({ \sqrt x})^2?

What are the test-makers thinking here?  I really don’t know.

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Bridges 2013 — Math and Art Conference

bridges 2013I am very excited to be heading to Enschede, the Netherlands later this week for the 2013 Bridges conference!

The Bridges organization has been hosting this international conference highlighting the connections between art, mathematics, and computer science for the past 15 years.  I have attended several Bridges conferences and have been greatly influenced by my experiences there.

This year I am excited to be exhibiting some work in the Bridges Mathematical Art Gallery.  You can see my pieces here, and browse the full galleries here.  I will also be presenting a short paper on some ideas about teaching mathematics through image manipulation, which relates to my pieces in the exhibition.

Bridges 2013 will be five days of inspiring people, conversations, mathematics, and art!  And after that, I’ll enjoy unpacking everything I experience throughout the school year.

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Regents Recap — June 2013: Solving Quadratic Equations

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

Solving equations is a fundamental mathematical skill, and it makes sense that we emphasize it in school curricula.  And since quadratic functions come up quite a bit in mathematical and scientific exploration, and offer a good balance of accessibility and complexity, it makes sense that solving quadratic equations is a particular point of emphasis.

This June, each of the three New York math Regents exams had at least one problem that required the student to solve a quadratic equation.  I don’t really have any objection to this, but what I find strange is the implied gap in mathematical content suggested by the types of questions asked.

Consider the following two questions.  The first is from the Integrated Algebra exam and the second is from the Algebra 2 / Trig exam.  These two exams, and their corresponding courses, are typically taken 2-3 years apart.

2013 June IA 17

2013 June A2T 36

The only difference between the content of these questions is the nature of the solutions of the equations.  In the first, the solutions are integers; in the second, the solutions are irrational numbers.  Thus, students are taught to solve quadratic equations with integer solutions in the Integrated Algebra course, but it isn’t until at least two years later that they are taught to solve quadratic equations with non-integer solutions.

That seems like an unreasonably long gap to me.  I’m not sure what the reasoning is behind waiting 2-3 years to teach students how to solve more complicated quadratic equations.  Maybe someone can make a sensible argument for this pacing and structure, but I’m not sure I can.

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