This simple exhibit at the New York Hall of Science demonstrates a lot of interesting trigonometry. As the light hits the boundary of the semi-circular glass block, it bends back toward the central axis of the system. I can’t look at this without seeing tangents, normals, and angles of incidence and refraction. I wish I had brought my protractor!
One of our end-of-school tasks was breaking down all of our Zometool builds from the past year. When a colleague handed me a bag of several hundred red, blue, and yellow struts, I immediately started to struggle and experiment with the most efficient ways to sort the struts for storage.
Should I take them out of the bag one at a time, placing each in its appropriate pile? Or maybe a handful of ten at a time? Or fifty? Ultimately I settled on dumping them out and sliding them around into color-coordinated piles. I’m not sure of the mathematical justification, but my internal optimizer seemed to think this was the right way to go.
To help our department prepare for the impending content shifts in our Algebra 2 course, I recently gave a demonstration lesson in probability and statistics. I was very lucky that my Skew Dice had just arrived!
Virtually everyone who encountered the skew dice had the same, immediate reaction: are the dice fair? This created an instant, authentic context for developing a wide variety of concepts and techniques in probability and statistics.
This simple question catalyzed natural mathematical conversations about what fairness means and how we might measure it. Transitioning from the intuitive notion that “each face should appear the same number of times” to a clear, rigorous mathematical characterization allowed us to wrestle with some fundamental statistical notions in a meaningful way.
I asked participants to propose tests for fairness, and then had them perform a test I had decided on ahead of time: roll the die 100 times and report the number of sixes. Before they began, I asked participants to consider how many sixes they would expect, and what numbers of observed sixes might suggest to them that the die was unfair.
The groups performed their tests and shared their data. We compared our results to our earlier intuitions, and talked about some ways we could interpret the data, touching on the rudiments of hypothesis testing.
A strength of this activity is that it creates opportunities to discuss modeling, experimental design, and data collection in meaningful ways: What assumptions did we make in our definitions of fairness? What assumptions underlie the test we conducted? What consequences follow from our choices about what data to collect, and how to collect it? All of these questions are interesting, important, and profoundly mathematical.
Another strength is that it engages participants in real mathematical inquiry, which I experienced firsthand when I performed the experiment myself. I ended up with an unusual number of 6s.
This prompted me to follow up with some more tests.
In the end, I felt confident with my conclusions, but the anomalous result had me reflecting on the process. As I thought about performing the test, I recalled frequently rolling the same number several times in a row. Luckily, the manner I chose to record the data allowed me to investigate how frequently I rolled consecutive numbers. The results were very surprising! This led me to ask, and contemplate, more questions about the skew dice. This is exactly the kind of mathematical experience I want students to have.
Skew dice are beautiful objects and great mathematical conversation starters. I highly recommend picking some up from The Dice Lab.
Much has been written and tweeted about this problem from a recent math exam administered in the UK:
After the exam, students took to social media to express their outrage at the absurdity of this question. This prompted some reaction pieces from mathematicians and math teachers defending this problem as a demonstration of a link between probability and algebra and as a non-routine problem-solving challenge.
The mathematical status of this problem is less interesting to me than its status as a test item. And as a test item, I think this is not only terrible, but also damaging.
The first eight sentences of this test item clearly indicate to the student that this is a probability problem. Then, it abruptly ceases to be a probability problem and becomes a problem about quadratic equations. No meaningful connection is made between the two concepts: the entire probability story simply exists to establish algebraic conditions on the number n. (And even in a world where contrived test questions are commonplace, this silly story stands out.)
For most students, this test question just reinforces the notion that math makes no sense. And I’m sure others come away feeling cheated, or deceived, by the exam-writers. High-stakes exam questions like this damage student attitudes about mathematics and learning, and have broad, long-term consequences that few people seem to think about.
This problem reminds me of the saga of “The Pineapple and the Hare“. A few years ago, a number of questions on an 8th-grade English exam referred to an absurd passage about a talking pineapple. The passage and the questions were published online, and the ensuing public outcry led to those items being nullified on the exam.
Yet test-writers defended the passage and the items as an effective discriminator: only the highest functioning test-takers could weave their way through the absurdity to answer the questions correctly. Thus, it effectively served to identify the highest performers.
Even if that were all true, why should the navigation of nonsense be a focus of our educational program? And what of the students who come away from such tests feeling demoralized and alienated because a probability abruptly became an algebra problem?
Regardless of what people think about this particular question, I’m glad that, more and more, we seem to be asking the question, “Are these tests any good?“.