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3D Printing in Math Class

We were fortunate to receive a 3D printer for use in our math class midway through the last school year.  Figuring out how it best worked was fun, and often frustrating.

We enjoyed a variety of successes throughout the spring, printing simple surfaces and some complicated ones, too.  It was fascinating to uncover how the printer, and its software and hardware, tackled certain engineering obstacles, like how to print in mid-air!

Ultimately I got comfortable enough to start producing some lesson-specific mathematical objects.  This trio of solids I designed worked perfectly as an introduction to Cavalieri’s principle:  seeing and holding the objects immediately initiated the conversations I wanted students to have.

By the end of the school year, I felt comfortable enough with the process to run our first official student project.  It was fairly open-ended, with options for students, but essentially the idea was to design an object for printing using equations and inequalities.

The project was a success, and here are some of the student designs.
Student 3d Prints

I’m looking forward to exploring some new ideas and projects this year.  It’s clear to me that this technology, which is fundamentally mathematical in concept and design, can play a valuable and meaningful role in math class.

Regents Recap — August 2015: Common Core Algebra

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

The August 2015 administration of the Common Core Algebra exam was similar in style, content, and difficulty to the prior Common Core Algebra exams.  There are a few interesting trends emerging.

Harder Multiple Choice

As part of the general increase in difficulty of these exams, we are seeing harder types of multiple choice questions.  Here are questions 9 and 21:

2015 August CC Alg 9

2015 August CC Alg 21

The simplest kind of multiple choice question in this style might just ask “Which of the following statements is true?”  It is generally more difficult to instead identify a false statement among a set, or to correctly identify the true or false subset of statements from a set.  I don’t object to these kinds of questions, but it’s worth acknowledging that this is one subtle way in which the difficultly of the exam can be tweaked.

Confusing Contexts

On each of the Common Core Algebra exams thus far, there have been real-world problems that I found very confusing.  In some cases, the more I read them, the less I understood what the question was asking.  Here is Question 18:

2015 August CC Alg 18

Apart from the decidedly unrealistic real-world context, I was quite confused about whether we were interested in monthly payments or total payments.  I wonder if these kinds of problems confuse students, or if they have just [properly] learned to ignore the model and just figure out what the test-maker wants to hear.


We have seen some Calculus-style content moved down into this Common Core Algebra exam.  Here, in Question 28, students are essentially asked to graph an antiderivative.

2015 August CC Alg 28I don’t have any philosophical objections to this particular content being part of an 8th- or 9th-grader’s mathematical experience, provided it’s part of a coherent curriculum.  But I do wonder about the inherent fairness of this as an 8th- or 9th-grade math exam question.

This question assesses an important concept in introductory physics.  Students in schools where Physics is taught in 9th grade will have a significant advantage on this kind of problem, while other students are in danger of being rated lower on mathematical proficiency simply because they haven’t taken physics.

This is another example of the virtually infinite set of confounding variables involved in assessing learning and teaching.

Teaching with “Why Do Americans Stink at Math?”

why do americans stink at mathMy latest piece for the New York Times Learning Network is a math lesson that uses Elizabeth Green’s article “Why Do Americans Stink at Math?” to get students thinking about the most effective ways to teach and learn mathematics.

Is there a crisis in math education? Lots of people seem to think so.

From worries about where the United States ranks on international tests to arguments over the Common Core, the way teachers teach and students learn math continues to be debated widely, leading to proposed changes in the ways mathematics is taught. But what really works for students in the math classroom? And when changes to the techniques are necessary, how can they be implemented effectively and appropriately across an entire system? This Text to Text lesson plan confronts those questions and more.

Students are invited to use the suggested texts, as well as their own experiences in math class, to explore questions like “Do you believe teaching with a stronger emphasis on conceptual understanding will improve students’ performance in math?”, “What are some of the potential obstacles one might face in trying to change the way mathematics, or any subject, is taught?”, and ultimately, “What are the best ways to teach and learn mathematics?”

The entire piece is freely available here.  There are already a number of interesting student comments on the piece.  It’s certainly eye-opening hearing what they have to say about how they perceive effective math teaching.

Math Photo: One-Fourteenth


How much blue do you see?  One-fourteenth is certainly an odd number to see around town.

9/12/15 — Happy Right Triangle Day!

Happy Right Triangle Day!  Today, on 9/12/15, we celebrate a favorite triangle:  the 9-12-15 right triangle.

9-12-15 Triangle

We know this triangle is right because the side lengths satisfy the Pythagorean Theorem.

9^2 + 12^2 = 15^2

It’s actually been less than a month since we last celebrated a Right Triangle Day, but they are rare, so enjoy being right today!