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…
I’ve been talking a lot with my colleagues about their teaching practices, as part of the NSF grant I’m working on. The inverted classroom (I used to call it the flipped classroom, but I’m going back to “inverted”) has come up a lot as a teaching technique that people have heard a lot about but haven’t tried yet — or are wary of trying. I’ve been wondering about the language being used, namely: Is the inverted classroom really a “teaching technique” at all?
My answer used to be “yes”. When I first started using the inverted classroom idea, I would describe the inverted classroom as “a teaching technique” that involves reversing where information transmission and internalization take place. Later I moved to saying that the inverted classroom refers to “any teaching method” that implements this reversal. Today as I was thinking about this, I think a…
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…
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…
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 …
Paul Pintrich was the creator of the Motivated Strategies for Learning Questionnaire, which I used as the main instrument for collecting data for the study on students in the flipped transition-to-proof course this past semester. Now that the data are in, I’ve been going back and reading some of Pintrich’s original papers on the MSLQ and its theoretical framework. What Pintrich has to say about student learning goes right to the heart of why I chose to experiment with the flipped classroom, and indeed I think he really speaks to the purpose of higher education in general.
For me, the main purpose of higher education is to train students on how to be learners — people who take initiative for learning things, who are skilled in learning new things, and who above all want to learn new things. My goal as an instructor is to make sure that every student in my class makes some form of…
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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