I have a confession to make: At this point in the semester (week 11), there’s a question I get that nearly drives me to despair. That question is:

Can we see more examples in class?

Why does this question bug me so much? It’s not because examples are bad. On the contrary, the research shows (and this is surely backed up by experience) that studying worked examples can be a highly effective strategy for learning a concept. So I ought to be happy to hear it, right?

When people ask this question because they want to study an example, I’m happy. But studying an example and seeing an example are two radically different things. Studying an example means making conscious efforts to examine the example in depth: isolating the main idea or strategy, actively trying out modifications to the objects involved, making connections to previous examples and mathematical results, and – very …

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 …

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….

Last week’s flare-up over Khan Academy was interesting on a number of levels, one of which is that we got a new look at some of the arguments used in KA’s favor. Perhaps one of the most prominent defenses against KA criticism is: Khan Academy is free and really helps a lot of people. You can’t argue with the “free” part. On the other hand, the part about “helping” is potentially a very strong argument in KA’s favor —but there are two big problems with the way in which this is being presented by KA people.

First, the evidence is almost entirely anecdotal. Look through the Pacific Research Institute whitepaper, for example, and the evidence presented in KA’s favor is anecdotes upon anecdotes — possibly compelling, but isolated and therefore no more convincing than the critics. The reason that anecdotes are not convincing is because for every anecdote that…

Dave (Coffey) sent me a tweet alerting me to this whitepaper published by the Pacific Research Institute, a free-market think tank based in San Francisco. “Look at page 14,” Dave said. I did, and found that I was being used as a prime example of a Khan Skeptic. Actually I am the last in a list of skeptics whose skepticism the authors attempt to dispatch. I’m in good company, as Keith Devlin is the first on that list and Veritasium…

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…

Elaine Seymour and Nancy Hewitt’s book Talking About Leaving: Why Undergraduates Leave the Sciences is considered one of the seminal works in the literature about STEM education in higher ed. It’s certainly one of the most cited. Even though it’s 15 years old, it still wields a powerful influence over a lot of thought about university-level STEM education.

Mark Connolly, a researcher at the Wisconsin Center for Education Research, recently reached out to me to make me aware that he and Anne-Barrie Hunter of the University of Colorado Boulder are conducting a follow-up study to re-evaluate one of the claims made in the original 1997 study by Seymour and Hewitt study. Mark asked me to post about this to the blog and solicit your help in conducting the study. This involves taking a two-question survey. Here is the announcement from Mark and Anne-Barrie, and I hope you can find the time…

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’m currently taking a MOOC called Computing for Data Analysis through Coursera. Ths is my fourth MOOC (the sixth one, if you count the two that I started and then dropped). It’s an introduction to the open-source statistical computing environment known as “R”. I got interested in R after learning about this modeling-based Calculus project that uses the statistical and plotting capabilities of R as well as some special symbolic packages as the centerpiece of introductory calculus. I’m leading a taskforce in my department to draft a plan for technology use in the Calculus sequence, and while I don’t think we’ll be using R, I like very much the spirit behind this calculus project, which puts programming at the heart of learning the subject and uses an open-source platform. Plus, I thought R might come in handy for analyzing my own data, and anyway, it’s free, and the course…

Here’s a piece of a conversation I just had with my 8-year old daughter, who is interested in becoming a teacher when she grows up.

Daughter: Dad, if you want to become a teacher, do you have to take classes?

Me: Yes. You have to take a lot of classes about how to teach and a lot of classes in the subjects you want to teach. You need to be really good at math to teach math, for example.

D: Then do you have to go out and teach in the schools, like Mr. D___ [the young man who student-taught in my daughter's elementary school this year]?

Me: That’s right. You have to take classes and you have to go into the schools and practice.

D: Do you have to practice with the little kids?

Me: That depends on who you want to teach. If you want to become an elementary school teacher you work with elementary school kids. If you want to teach in a middle school, then you work with middle …

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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