Tag Archives: mathematics

March 18, 2014, 4:34 pm

What should mathematics majors know about computing, and when should they know it?

5064804_8d77e0d256_mYesterday I got an email from a reader who had read this post called What should math majors know about computing? from 2007. In the original article, I gave a list of what computing skills mathematics majors should learn and when they should learn them. The person emailing me was wondering if I had any updates on that list or any new ideas, seven years on from writing the article.

If anything, over the past seven years, my feelings about the centrality of computing in the mathematics major have gotten even more entrenched. Mostly this is because of two things.

First, I know more computer science and computer programming now than I did in 1997. I’ve learned Python over the last three years along with some of its related systems like NumPy and SciPy, and I’ve successfully used Python as a tool in my research. I’ve taken a MOOC on algorithms and read, in whole or in part, books…

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July 23, 2013, 8:00 am

Doubt, proof, and what it means to do mathematics

8426562723_d5b0d9b0ec_nYesterday I was doing some literature review for an article I’m writing about my inverted transition-to-proof class, and I got around to reading a paper by Guershon Harel and Larry Sowder¹ about student conceptions of proof. Early in the paper, the authors wrote the following passage about mathematical proof to set up their main research questions. This totally stopped me in my tracks, for reasons I’ll explain below. All emphases are in the original.

An observation can be conceived of by the individual as either a conjecture or as a fact.

A conjecture is an observation made by a person who has doubts about its truth. A person’s observation ceases to be a conjecture and becomes a fact in her or his view once the person becomes certain of its truth.

This is the basis for our definition of the process of proving:

By “proving” we mean the process employed by an…

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March 27, 2013, 8:00 am

Inside the inverted transition-to-proofs class: What the students said

6799510744_d2085426d7_mIn my series of posts on the flipped intro-to-proofs course, I’ve described the ins and outs of the design challenges of the course and how the course was run to address those challenges and the learning objectives. There’s really only one thing left to describe: How the course actually played out through the semester, and especially how the students responded.

I wasn’t sure how students in the course would respond to the inverted classroom structure. On the one hand, by setting the course up so that students were getting time and support on the hardest tasks in the course and optimizing the cognitive load outside of class, this was going to make a problematic course very doable for students. On the other hand, students might be so wed to the traditional classroom setup that no amount of logic was going to prevail, and it would end up like my inverted MATLAB class did where a

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March 18, 2013, 8:00 am

Inside the inverted proofs class: What we did in class

3095855157_7bf2df04fa_mI’ve written about the instructional design behind the inverted transition-to-proofs course and the importance of Guided Practice in helping students get the most out of their preparation. Now it comes time to discuss what we actually did in class, having freed up all that time by having reading and viewing done outside of class. I wrote a blog post in the middle of the course describing this to some degree, but looking back on the semester gives a slightly different picture.

As I wrote before, each 50-minute class meeting was split up into a 5-minute clicker quiz over the reading and the viewing followed by a Q&A session over whatever we needed to talk about. The material for the Q&A was a combination of student questions from the Guided Practice, trends of misconceptions that I noticed in the Guided Practice responses (whether or not students brought them up), quiz questions with…

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March 13, 2013, 8:00 am

Inside the inverted proofs class: Guided Practice holds it together

In the last couple of posts on the inverted transition-to-proofs course, I talked about course design, and in the last post one of the prominent components of the course was an assignment type that I called Guided Practice. In my opinion Guided Practice is the glue that held the course together and the engine that drove it forward, and without it the course would have gone a little like this.

So, what is this Guided Practice of which I speak?

First let’s recall one of the most common questions asked by people learning about the inverted classroom. The inverted classroom places a high priority on students preparing for class through a combination of reading, videos, and other contact with information. The question that gets asked is — How do you make sure your students do the reading? Well, first of all I should say that the answer is that there really is no simple way to …

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March 11, 2013, 8:00 am

Inside the inverted proofs class: Meeting the design challenges

2688582584_644b85622e_nIt’s been a while since I last wrote about the recently-completed inverted transition-to-proof course. In the last post, I wrote about some of the instructional design challenges inherent in that course. Here I want to write about the design itself and how I tried to address those challenges.

To review, the challenges in designing this course include:

  • An incredibly diverse set of instructional objectives, including mastery of a wide variety new mathematical content, improvement in student writing skills, and metacognitive objectives for success in subsequent proof-based courses.
  • The cultural shock encountered by many students when moving from a procedure-oriented approach to mathematics (Calculus) to a conceptual approach (proofs).
  • The need for strong mathematical rigor, so as to prepare students well for 300-level proof based courses, balanced with a concern for student…

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July 10, 2012, 11:38 am

Inverting the transition-to-proofs class

When I see the first back-to-school sales, I know it’s time, like it or not, to start prepping classes for the fall. This fall I am teaching two courses: a second-semester discrete math course for computer science majors and then two sections of “Communicating in Mathematics” (MTH 210). I’ve written about MTH 210 before when I taught it last fall. This fall, it’s going to be rather different, because I’m designing my sections as inverted or “flipped” classes.

If you’ve read this blog for any length of time, you know I’ve worked with the inverted classroom before (here, here, here, etc.). But except for a few test cases, I haven’t done anything with this design since coming to GVSU. I decided to take a year off from doing anything inverted last year so I could get to know the students and the courses at GVSU and how everything fits together. But now that I have the lay of the land, I…

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June 5, 2012, 8:00 am

Taking time and giving time on assessments

I haven’t given many updates lately about, well, anything, but especially about my Calculus 2 class. Freakishly, we are 2/3 of the way through the course now. First of all let me say that there’s something seriously wrong with having a midterm in a class after three weeks, and then a final exam three weeks later. Students should have more time to dread those things.

I kid, but actually the biggest adjustment I’ve made in the class — and teaching a class that’s as compressed as this one is all about paying close attention to everything that happens and being nimble about making adjustments — has been the testing scheme. I know that I posted earlier about my idea of having in-class assessments that were smaller than the usual test, more frequent, and which leveraged student collaboration and the real-life social network of the class. But after a couple of tries with this, I dropped it…

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April 12, 2012, 9:45 am

Can Math Be Made Fun?



This article in the Chicago Tribune talks about efforts to make math fun:

In the American drive to boost science and math education, it’s science that has all the kid-friendly sizzle: Robots and roller coasters, foaming chemical reactions, marshmallow air cannons.

Math has… well, numbers.

“America has a cultural problem with math. It’s the subject, more than any other, that we as a country love to hate,” said Glen Whitney, a passionate mathematician who worked for years developing algorithms for hedge funds. “We don’t see it as dynamic. It’s rote and boring and done by dead Greek guys a thousand years ago.”

The article goes on to talk about some efforts to spice up math, including MIT’s Labyrinth tournament, DimensionU‘s celebrity-driven “DU the Math”…

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February 20, 2012, 7:32 pm

The origin of the “nabla” symbol

We’re about to start working with gradient vectors in Calculus 3, and this topic uses a curious mathematical symbol: the nabla, which looks like: \(\nabla\). This symbol has several mathematical uses, one of which is for gradients; if \( f  \) is a function of two or more variables then \( \nabla f \) is its gradient. But there does not appear to be a use for the symbol outside mathematics (and mathematical physics).

One of my students asked me about the origin of this symbol, and I had to confess I didn’t know. I always figured it was somehow related to the much more common capital Greek delta, \( \Delta \), but the real story is a lot more colorful than that.

The nabla is so-called because it looks like a harp; the Greek word for the Hebrew or Egyptian form of a harp is “nabla” . What does a harp have to do with mathematics? The image came up in relation to mathematics…

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