Category Archives: Problem Solving

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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June 25, 2013, 12:04 pm

What math topic do engineering faculty rate as the most important?

did-an-even-math-problem-by-accident-answer-not-in-back-of-bookI’m at the American Society for Engineering Education Annual Conference right now through Thursday, not presenting this time but keeping the plates spinning as Mathematics Division program chair. This morning’s technical session featured a very interesting talk from Kathy Harper of the Ohio State University. Kathy’s talk, “First Steps in Strengthening the Connections Between Mathematics and Engineering”, was representative of all the talks in this session, but hers focused on a particular set of interesting data: What engineering faculty perceive as the most important mathematics topics for their areas, and the level of competence at which they perceive students to be functioning in those topics.

In Kathy’s study, 77 engineering faculty at OSU responded to a survey that asked them to rate the importance of various mathematical topics on a 5-point scale, with 5 being the…

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May 8, 2013, 8:00 am

Is the modified Moore method an instance of the flipped classroom?

I was really fortunate this past weekend to host Dana Ernst and T.J. Hitchman, two colleagues (from Northern Arizona University and University of Northern Iowa, respectively) at the Michigan MAA section meeting. They did a discussion panel on Teaching to Improve Student Learning that I organized, and we ended talking a lot about inquiry-based learning, which both of these guys practice. After Dana blogged about the session, he got this tweet:

Dana, Brandon, and I exchanged some tweets after that, and I think generally we’re on the same page, but here’s my reasoning about this question and, more generally, what does or does not fall under the heading of “flipped classroom”.

The main thing to keep in mind is the distinction between an instructional practice and a course design principle. This was the gist of my post a…

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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 20, 2013, 8:00 am

Inside the inverted proofs class: Dealing with grading

6251461402_2a11c771db_mSo, what about grading in that inverted transition-to-proofs course? Other than the midterm and final exams, which were graded pretty much as you might expect, we had four recurring assignments that required grading: Guided Practice, Quizzes, Classwork, and the Proof Portfolio. Let’s discuss the workflow and how it was all managed.

Let’s start with the easy stuff: Quizzes and Guided Practice. Quizzes were done using clickers, so the grading was trivial. Guided Practice was graded on the basis of completeness and effort only, on a scale of 0–2. So it was almost instantaneous to grade. Students would submit their work using a Google form that dumped their responses into a spreadsheet. I would just sort the spreadsheet in alphabetical order, look through for any glaring omissions or places where effort was lacking, and then put the grades right into Blackboard. A grade of “0”…

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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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January 28, 2013, 7:45 am

Inside the inverted proofs class: Design challenges

This is the second post in a series on the nuts and bolts behind the inverted transition-to-proofs course. The first post addressed the reasons why I decided to turn the course from quasi-inverted to fully inverted. Over the next two posts, I’m going to get into the design of the course and some of the principles I kept in mind both before and during the semester to help make the course work. Here I want to talk about some of the design challenges we face when thinking about MTH 210.

As with most courses, I wanted to begin with the end in mind. Before the semester begins, when I think about how the semester will end, the basic questions for me are: What do I want students to be able to do, and how should they be doing it?

This course has a fairly well defined, standard set of objectives, all centered around using logic and writing mathematical proofs. I made up this list that has…

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