Eigenightmares, Dancing Stick Figures, and the Advantages of a Spiral Approach to Pedagogy

For my next installment on innovative teaching techniques, I’d like to dredge another demon that haunted my nights long ago—EIGENVECTORS!  Normally, eigenvectors are introduced in the waning days of a first-year linear algebra course, when students’ minds are already saturated with new material and the stress of the later weeks of the semester, leading some to blindly memorize their theorems without really gaining a deep appreciation and understanding of them.  Hence my recurring “eigenightmares” of sophomore year.  One way around that pitfall is to introduce eigenvectors early on and then circle back to them throughout the course so students have a chance to use them again and again in a variety of contexts.  This pedagogical strategy, known the spiral approach, can be applied to almost any skill or concept you wish to teach.

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Getting Students Used to Abstract Concepts

According to my experience teaching limits in Calculus for Life Scientists, it is really difficult for students to understand limits because they do not have the required knowledge on which to build the concept. I have noticed that even if they can handle and apply the limit definition, problems often arise with their supposed prerequisite knowledge such as simplification, least common denominators, factoring, and long division. I had to make sure to spend the first two weeks of classes reviewing and refreshing what they had forgotten from their high school math classes.

The function machine by Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License (http://creativecommons.org/licenses/by-nc-sa/4.0/).

It can take a while for students to become comfortable with the idea and practice of treating functions as abstract objects. The function machine by Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License (http://creativecommons.org/licenses/by-nc-sa/4.0/).

However, after introducing limits, the students still encountered difficulties understanding the subtleties of one-sided versus two-sided limits. When you explain how to handle some types of functions, students will Continue reading

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Graduate School: Where Grades Don’t Matter

 Guest post by Tai-Danae Bradley

Yesterday I received a disheartening 44/50 on a homework assignment. Okay okay, I know. 88% isn’t bad, but I had turned in my solutions with so much confidence that admittedly, my heart dropped a little (okay, a lot!) when I received the grade. But I quickly had to remind myself, Hey! Grades don’t matter.

PhD Comic Tai

“Piled Higher and Deeper” by Jorge Cham www.PhDComics.com, used with permission.  Annotations by Tai-Danae Bradley.

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Pixar in a Box: My New Answer to “What’s this good for?”

Questions like “Why do we need to know this?” or “When am I ever going to use this in real life?” seem to show up every time I put the definition of the derivative on the board in a calculus class or work through the different ways to express a line with some algebra students. We’ve all probably even asked these questions ourselves at various points in our mathematical careers. Every time I try to answer these questions, I give a few quick examples, but I never feel like I’ve really convinced anyone how cool it is.

These conversations are always short, and I have to remind myself that I didn’t fall in love with math because someone told me that I could use it to build bridges or maximize profit. I love math, because I’ve spent time seeing what math can do and doing it myself.

Using combinations to build robots! Photo by Sarah Salmon.

Using combinatorics to build robots on Khan Academy! Photo by Sarah Salmon.

Now, I finally have a better answer. Last August, Khan Academy and Pixar teamed up to release Pixar in a Box!

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Computing π with Billiards (Mathematical Puzzles/Riddles, Part IV)

In honor of this past Monday being Pi day, I thought I would post a special mathematical oddity/riddle related to calculating π. Many of you may be familiar with the famous Buffon’s needle problem, in which π is estimated by throwing needles on the floor and using some elementary calculus and probability theory. Here we will describe another amusing method for computing π (perhaps not very practical), involving some basic physics!

COMPUTING PI VIA BILLIARDS

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