Category Archives: Problem Solving

January 22, 2013, 8:00 am

Inside the inverted proofs class: Why I did it

It’s been a month or so now that the inverted transition-to-proofs class drew to a close. A lot of people, both here at my institution and online, have been asking questions about the design and day-to-day operations of the course, especially if they have ideas of their own and want to compare notes. So starting with this post, I’m going to publish a series of posts that describe exactly how this course was designed and managed throughout the semester. I’m not sure how many of these posts there will be. But the idea is to pull everything together so that people who want to try this sort of thing themselves will have a detailed accounting of what I did, what worked, what didn’t, and how it all went.

Some background on the course (MTH 210: Communicating in Mathematics) is in this post. The short version is that MTH 210 is a course on reading and writing proofs. It’s a…

September 10, 2012, 3:13 pm

How do student attitudes toward learning change in a flipped classroom?

Sorry for the absence, but things have been busy around here as we step fully into the new semester. The big experiment this term is with my flipped introduction to proofs class. As I wrote last time, I was pretty nervous going into the semester about the course. But things seem to be working really well so far. I don’t want to jinx the experience by saying so, but so far, nobody in either of my two sections of the course has given any indication that the flipped model isn’t working for them. In fact, I gave a survey in the first week of class that included an item soliticing their concerns or questions about the flipped model, and here’s a sample of the responses:

• I think the “flipped” structure will be better for a lot of the students and end with success from more students than normal.
• I think this sounds really great. The idea of actually working on problems in class …

August 14, 2012, 8:00 am

How the inverted classroom works: A manifesto for students

I’ve been sort of quiet on the inverted transition-to-proof course (MTH 210, Communicating in Mathematics) lately, partly due to MathFest and partly because I am having to actually prep said course for startup on August 27. It’s almost ready for launch, and I wanted to share a document that I’m going to hand out to students on opening day and discuss. It’s called “How MTH 210 Works”. I’m fairly proud of this document because I think it says, in clear terms, what I want students to know not only about this class but for inverted classrooms generally.

I’ve written before that the inverted or “flipped” classroom approach always tends to engender a lot of uncertainty and sometimes strongly negative responses. With this document, I am hoping to pre-empt a lot of those feelings by stressing what this is all about: Being realistic about their education in the present day for the things that…

July 18, 2012, 9:44 am

For proofs, just click “play”

This week I am adding to the playlist of screencasts for the inverted intro-to-proofs class I first mentioned here. There are seven chapters in the textbook we are using and my goal is to complete the screencasts for the first three of those chapters prior to the start of the semester (August 27). Yesterday I added four more videos and I am hoping to make four more tomorrow, which will get us through Chapter 1.

The four new ones focus on conditional (“if-then”) statements.  I made this video as the second video in the series as a prelude to proofs, which are coming in Section 1.2 and which will remain the focus of the course throughout. Generally speaking, students coming into this course have had absolutely no exposure to proof in their background with the exception of geometry and maybe trigonometry, in which they hated proofs. Watch a part of this and see if you can figure out my …

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…

June 12, 2012, 7:00 am

Misunderstandings vs. misconceptions

The first speaker in the Model-Eliciting Activities (MEA’s) session Monday morning said something that I’m still chewing on:

Misunderstanding is easier to correct than misconception.

She was referring to the results of her project, which took the usual framework for MEA’s and added a confidence level response item to student work. So students would work on their project, build their model, and when they were done, give a self-ranking of the confidence they had in their solution. When you found high confidence levels on wrong answers, the speaker noted, you’ve uncovered a deep-seated misconception.

I didn’t have time, but I wanted to ask what she felt the difference was between a misunderstanding and a misconception. My own answer to that question, which seemed to fit what she was saying in the talk, is that a misunderstanding is something like an incorrect interpretation of an idea …

February 27, 2012, 8:00 am

Encountering abstraction with clickers

I had the great pleasure this weekend of leading a session at Math In Action, which is Grand Valley’s annual K-12 educators’ conference. My session was called “Classroom Response Systems in Mathematics: Learning math better through voting” and was all about the kinds of learning that can take place in a class where active student choice is central and clickers are mediating the voting. (Here are the slides.)

It always seems like a bait-and-switch when I do a “clicker” workshop, because although people come to learn about clickers, I don’t really have much to say about the technology itself. As devices go, clickers are about as complex as a garage door opener, and in fact they work on the same principle. There’s not a lot to discuss. So instead, we spend our time focusing on the kinds of pedagogy that clickers enable — which tends to excite teachers more than technology does.

The a…

September 15, 2011, 8:52 pm

Using clickers for peer review of proofs

http://www.flickr.com/photos/unav/

Right now I’m teaching a course called Communicating in Mathematics, which serves two purposes. First, it’s a transitional course for students heading from the freshman calculus sequence into more theoretical upper-level math courses. We learn about logic, how to formulate and test mathematical conjectures, and we spend a lot of time learning how to write correct mathematical proofs. And therein is the second purpose: The course is also labelled as a “Supplemental Writing Skills” course at Grand Valley, which means that a large portion of the class, and of the course grade, is based on writing. (Here are the specifics.) It’s a sort of second-semester, discipline-specific composition class. (Students at GVSU have to have two of these SWS courses, each in different…

May 26, 2011, 8:01 pm

As part of preparing for our impending move from Indy to Grand Rapids, my family and I have made a couple of visits to the area. These by necessity combine business with pleasure, since our three kids (ages 2, 5, and 7) don’t handle extended amounts of business well. On the last visit, we spent some time at the Grand Rapids Childrens Museum, the second floor of which is full of stuff that could occupy children — and mathematicians — for hours. This “exhibit” was, for me, one of the most evocative. Have a look:

I asked this on Twitter a few days ago, but I’ll repost it here: In the spirit of Dan Meyer’s Any Questions? meme, what questions come to mind as you watch this? Particularly math, physics, etc. questions.

One other thing — just after I wrapped up the video on this, someone put one of the little discs rolling on the turntable…

December 16, 2010, 2:30 pm

A problem with "problems"

I have a bone to pick with problems like the following, which is taken from a major university-level calculus textbook. Read it, and see if you can figure out what I mean.

This is located in the latter one-fourth of a review set for the chapter on integration. Its position in the set suggests it is less routine, less rote than one of the early problems. But what’s wrong with this problem is that it’s not a problem at all. It’s an exercise. The difference between the two is enormous. To risk oversimplifying, in an exercise, the person doing the exercise knows exactly what to do at the very beginning to obtain the information being requested. In a problem, the person doesn’t. What makes an exercise an exercise is its familiarity and congruity with prior exercises. What makes a problem a problem is the lack of these things.

The above is not a problem, it is an exercise. Use the

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