Why we need Receptive Learning to have Active Learning

Posted on October 5, 2017 by James Sheldon

In a recent issue of Notices of the AMS, Benjamin Braun, Priscilla Bremser, Art M. Duval, Elise Lockwood, and Diana White make a compelling case to include active learning in mathematics. I want to make a less popular move and ask, what is so bad about the flip side of active learning, or in other words, what’s so bad about receptive learning?

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Applying to grad school? Here’s what you need to know: Part I

Posted on September 29, 2017 by Susannah Shoemaker

I put together my grad school applications when I was studying abroad in St. Petersburg, Russia, which meant that they were cobbled together in a string of internet cafes, fueled by little more than espresso and impatience. I dashed off emails to professors in hip anti-cafes with unmarked doors and killer espresso, asking for recommendation letters and moral support. I wrote my statement of purpose over weak instant coffee (but fast Wi-Fi!) on the top floor of a mall. While I love a coffee-shop crawl as much as the next person, I was anxious to have my applications out of the way so that I could explore St. Petersburg and get to know its quirks and its grandeur.

And I did. The circumstances of my application-writing meant that I was efficient: I got things out of the way as early as I could, and as a result, I wasn’t racing to meet deadlines come December – instead, I was out and about, exploring my temporary home. On the other hand, I was not as thoughtful as I might have been in putting together my applications, and in considering what my grad school experience might look like.

While I relish the memories of my whirlwind semester abroad, here’s some advice to make your grad school application process a little less hectic­­ ­– and a little more organized ­– than mine. This is the first part of a two-part post; Part II will be posted next month. Continue reading →

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Real Numbers Base…Factorials! And A By-product

Posted on September 20, 2017 by Behnam Esmayli

PROPOSITION 1:  For a real number  x  there exists a sequence $ x_1, x_2, x_3,…$ of integers such that

$ \hspace{4cm} x=x_1 +\frac{x_2}{2!}+\frac{x_3}{3!} + \cdots + \frac{x_n}{n!} + \cdots,  \hspace{2cm} (*) $

where $x_1$ can be any integer, but for $ n \geq 2$, $x_n \in \{ 0,1,…,n-1 \}.$ Furthermore, if we require that the partial sums be strictly smaller than  x, then such a representation is unique.

Remark: One cannot help recalling decimal or binary expansion of numbers. Notice that $\frac{n}{n!}=\frac{1}{(n-1)!}$ (drops back to previous digit), so the bound on $x_n$ is logical. Continue reading →

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Daily Quizzes: the Good, the Bad, and the Ugly—Part 2

Posted on September 11, 2017 by Sarah K. Salmon

You may recall that quite some time ago, I tried to convince you that giving your students a one- or two-question quiz every single day had a myriad of good aspects. You can check out why I loved this method in Part 1. As a quick refresher, I taught Calculus I four days a week the semester that I employed this method. Now, we’re going to discuss the bad (easily fixable) and ugly (not so easily fixable) issues which I ran into that semester. To keep this post from being a total downer, we are also going to talk about a new experiment I tried the next semester that I taught.  Continue reading →

Posted in Math Education, Teaching | Tagged , , | 3 Comments

Shedding light on AI’s black boxes

Posted on September 3, 2017 by Susannah Shoemaker

A recent special issue in Science highlights the increasingly important role that artificial intelligence (AI) plays in science and society. Providing a small but compelling sample of the types of challenges AI is equipped to tackle—from aiding chemical synthesis efforts to detecting strong gravitational lenses—the issue captures the palpable excitement about AI’s potential in a world saturated with data.

But one article in particular, “The AI detectives,” captured my attention. Rather than highlighting a specific application of AI, as the other articles do, this piece draws attention to the lack of transparency in certain machine learning algorithms, particularly neural networks. The inner workings of such algorithms remain almost entirely opaque, and they are accordingly termed “black boxes”: though they may generate accurate results, it’s still unclear how and why they make the decisions they do.

Machine Learning by XKCD is licensed under CC BY 2.5.

Researchers have recently turned their attention to this problem, seeking to understand the way these algorithms operate. “The AI detectives” introduces us to these researchers, and to their approaches to unlocking AI’s black boxes. Continue reading →

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