The semester just ended, and I’m now in full retrospect mode. This semester I was fortunate to have only one prep — two sections of Linear Algebra. Linear algebra, for me, is the cornerstone of a modern mathematics education precisely because its concepts and its mechanics lie at the heart of so much real-world stuff — from web search algorithms to scheduling problems to computer graphics and many other areas. And yet, in a typical one-semester course on linear algebra you only get to touch on a handful of applications, and those tend to be sort of domesticated. A few years ago, I decided I wanted students to explore more than just the stock examples in the textbook, and I wanted them to do so in an authentic way that reflects real-world mathematical practice.

Screencasting is an integral part of the inverted classroom movement, and you can find screencasting even among courses that aren’t truly flipped. Using cheap, accessible tools for making and sharing video to clear out time for more student-active work during class make screencasting very appealing. But does it work? Do screencasts actually help students learn?

We have lots of anecdotal evidence that suggests it does, but it turns out there are actually data as well that point in this direction. I’ve been reading an article by Katie Green, Tershia Pinder-Grover, and Joanna Mirecki Millunchick (of Michigan State University and the University of Michigan) from the October 2012 issue of the Journal of Engineering Education in which they studied 262 students enrolled in an engineering survey course that was augmented with screencasts. Here’s the PDF. This paper is full of interesting…

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

So, 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”…

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…

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 …

It’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…

Linear algebra is a strange course in some ways. There are a lot of mechanical skills one has to learn, like multiplying matrices and performing the Row Reduction Algorithm. If you come into linear algebra straight out of calculus with a purely instrumental viewpoint on mathematics, you will almost certainly think that these mechanical skills are the point of linear algebra. But you’d be wrong! It’s the conceptual content of the subject that really matters. Like I tell my students, you can answer almost any question in linear algebra by forming a matrix and getting it to reduced row echelon form….

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…

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…

I am a mathematician and educator with interests in cryptology, computer science, and STEM education. I am affiliated with the Mathematics Department at Grand Valley State University in Allendale, Michigan. The views here are my own and are not necessarily shared by GVSU.

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