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Targeting the inverted classroom approach

Eigenvector

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A while back I wondered out loud whether it was possible to implement the inverted or “flipped” classroom in a targeted way. Can you invert the classroom for some portions of a course and keep it “normal” for others? Or does inverting the classroom have to be all-or-nothing if it is to work at all? After reading the comments on that piece, I began to think that the targeted approach could work if you handled it right. So I gave it a shot in my linear algebra class (that is coming to a close this week).

The grades in the class come primarily from in-class assessments and take-home assessments. The former are like regular tests and the latter are more like take-home tests with limited collaboration. We had online homework through WeBWorK but otherwise I assigned practice exercises from the book but didn’t take them up. The mix of timed and untimed assessments worked well enough, but the lack of collected homework was not giving us good results. I think the students tended to see the take-home assessments as being the homework, and the WeBWorK and practice problems were just something to look at.

What seemed true to me was that, in order for a targeted inverted classroom approach to work, it has to be packaged differently and carry the weight of significant credit or points in the class. I’ve tried this approach before in other classes but just giving students reading or videos to watch and telling them we’d be doing activities in class rather than a lecture — even assigning  minor credit value to the in-class activity — and you can guess what happened: nobody watched the videos or read the material. The inverted approach didn’t seem different enough to the students to warrant any change in their behaviors toward the class.

So in the linear algebra class, I looked ahead at the course schedule and saw there were at least three points in the class where we were dealing with material that seemed very well-suited to an inverted approach: determinants, eigenvalues and eigenvectors, and inner products. These work well because they start very algorithmically but lead to fairly deep conceptual ideas once the algorithms are over. The out-of-class portions of the inverted approach, where the ball is in the students’ court, can focus on getting the algorithm figured out and getting a taste of the bigger ideas; then the in-class portion can focus on the big ideas. This seems to put the different pieces of the material in the right context — algorithmic stuff in the hands of students, where it plays to their strengths (doing calculations) and conceptual stuff neither in a lecture nor in isolated homework experiences but rather in collaborative work guided by the professor.

To solve the problem of making this approach seem different enough to students, I just stole a page from the sciences and called them “workshops“. In preparation for these three workshops, students needed to watch some videos or read portions of their textbooks and then work through several guided practice exercises to help them meet some baseline competencies they will need before the class meeting. Then, in the class meeting, there would be a five-point quiz taken using clickers over the basic competencies, followed by a set of in-class problems that were done in pairs. A rough draft of work on each of the in-class problems was required at the end of the class meeting, and students were given a couple of days to finish off the final drafts outside of class. The whole package — guided practice, quiz, rough draft, and final draft — counted as a fairly large in-class assessment.

Of course this is precisely what I did every week in the MATLAB course. The only difference is that this is the only way we did things in the MATLAB course. In linear algebra this accounted for three days of class total.

Here are the materials for the workshops we did. The “overview” for each contains a synopsis of the workshop, a list of videos and reading to be done before class, and the guided practice exercises.

The results were really positive. Students really enjoyed doing things this way — it’s way more engaging than a lecture and there is a lot more support than just turning the students out of class to do homework on their own. As you can see, many of the guided practice exercises were just exercises from the textbook — the things I had assigned before but not taken up, only to have them not done at all. Performance on the in-class and take-home assessments went up significantly after introducing workshops.
Additionally, we have three mastery exams that students have to pass with 100% during the course — one on row-reduction, another on matrix operations, and another on determinants. Although determinants form the newest and in some ways the most complex material of these exams, right now that exam has the highest passing rate of the three, and I credit a lot of that to the workshop experience.
So I think the answer to the question “Can the inverted classroom be done in a targeted way?” is YES, provided that:
  • The inverted approach is used in distinct graded assignments that are made to look and feel very distinct from other elements of the course.
  • Teachers make the expectations for out-of-class student work clear by giving an unambiguous list of competencies prior to the out-of-class work.
  • Quality video or reading material is found and used, and not too much of it is assigned. Here, the importance of choosing a textbook — if you must do so — is very important. You have to be able to trust that students can read their books for comprehension on their own outside of class. If not, don’t get the book. I used David Lay’s excellent textbook, plus a mix of Khan Academy videos and my own screencasts.
  • Guided practice exercises are selected so that students experience early success when grappling with the material out of class. Again, textbook selection should be made along those lines.
  • In-class problems are interesting, tied directly to the competency lists and the guided practice, and are doable within a reasonable time frame.
These would serve as guidelines for any inverted classroom approach, but they are especially important for making sure that student learning is as great or greater than the traditional approach — and again, the idea of distinctness seems to be the key for doing this in a targeted way.
What are your suggestions or experiences about using the inverted or “flipped” classroom in a targeted way like this?
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Filed under Clickers, Inverted classroom, Linear algebra, MATLAB, Screencasts

Calculus reform’s next wave

There’s a discussion going on right now in the Project NExT email list about calculus textbooks, the merits/demerits of the Stewart Calculus textbook, and where — if anywhere — the “next wave” of calculus reform is going to come from. I wrote the following post to the group, and I thought it would serve double-duty fairly well as a blog post. So… here it is:

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I’d like to add my $0.02 worth to this discussion just because (1) I’m a longtime Stewart Calculus user, having used the first edition (!) when I was an undergrad and having taught out of it for my entire career, and (2) I’m also a fairly consistent critic of Stewart’s calculus and of textbooks in general.

I try to see textbooks from the viewpoints of my students. From that vantage point, I unfortunately find very little to say in favor of Stewart’s franchise of  books, including the current edition, all of the previous five editions, the CCC version (which is almost exactly the same as the non-reform version of the book but with less clarity in its language), or the “Essential” calculus edition. Stewart has a relentlessly formalistic approach to calculus that, while admirable in its rigor, renders it all but impenetrable to students who are not used to such an approach, which is certainly nearly every student I teach and I would imagine a large portion of the entire population of beginning calculus students.

If you don’t believe me, go check out his introductory section on the definite integral (Section 5.2 in the sixth edition). Stewart hopelessly confuses the essentially very simple idea of the definite integral by hitting students with an avalanche of sigma-notation right out of the gate. Or, try the section on exponential functions (1.5), in which Stewart for some reason feels like it’s necessary to explain how it is we can define an exponential function at rational and irrational inputs. This is all well and good, but does the rank-and-file beginning calculus student need to know this stuff, right now?

As a result, I find myself having to tell students NOT to read certain portions of the book, and then remixing and rewriting large parts of the rest of it. But that leads to the ONE thing I can say in the positive sense about Stewart, which I can’t say about many “reform” books: Stewart is what you make it. The book does not force me to teach in a certain way, and if I want to totally ignore certain parts of it and write my own stuff, then this generally doesn’t cause problems down the road. For example, at my college we don’t cover trigonometry in the first semester. In most other books we’ve examined, trig and calculus are inextricable, and so the books are unusable for us. With Stewart, though, given a judicious choice of exercises to omit, you can actually pull off a no-trig Calculus I course with very little extra work on the prof’s part.

I can also say, regarding Stewart CCC, that the ancillary materials are excellent. The big binder of group exercises that comes with the instructor edition is much better than the book itself.

I don’t think that I have yet seen a calculus book that is really fundamentally different from the entire corpus of calculus textbooks, with the possible exception of Hughes & Hallett. They all cover the same topics in the same order, more or less, and in the same ways. If you’re looking for the next wave of calculus reform, therefore, you’ll have to find it outside the confines of a textbook, or at least the textbooks that are currently on the market. Textbooks almost by definition are antithetical to reform. Perhaps real reform will come with the rejection of textbooks as authoritative oracles on the subject in the first place. That could mean designing courses with no centralized information source, or using “inverted classroom” models utilizing online resources like the videos at Khan Academy (http://www.khanacademy.org) or iTunesU, or some combination of these.

Actually, more likely the next wave of reform will be in the form of reconsidering the place of calculus altogether, as the CUPM project did several years ago. Is it perhaps time to think about replacing calculus with a linear combination (pardon the pun) of statistics, discrete math, and linear algebra as the freshman introduction to college mathematics, or at least letting students choose between calculus and this stat/discrete/linear track? Is calculus really the best possible course for freshmen to take? I think that’s a discussion worth having, or reopening.

Enjoy,
Robert Talbert
Franklin College
Peach Dot 1997-1998

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Filed under Calculus, Education, Linear algebra, Math, Teaching, Textbook-free, Textbooks

A business model for free content

In a comment on an earlier post, I said I would try to blog about Flat World Knowledge and their business model soon. Here’s a 20-minute video that goes over this business model which allows textbooks to be free but still provides compensation to authors.

Vodpod videos no longer available.

more about “A business model for free content“, posted with vodpod

Again: Free textbooks can be done; it just requires a different approach than the one we’re used to.

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Free textbooks: It can be done

The last time I taught abstract algebra, I used no textbook but rather my own homemade notes. That went reasonably well, but in doing initial preps for teaching the course again this coming fall I realized my notes needed a serious overhaul; and since I’m playing stay-at-home dad to three kids under 6 this summer, this is looking more like a sabbatical project than something I can get done before August. So last month I set about auditioning textbooks.

I looked at the usual suspects — the excellent book by Joe Gallian which I’ve used before and really liked, Hungerford’s undergraduate text*, Rotman — but in the end,  I went with Abstract Algebra: Theory and Applications by Tom Judson. I would say it’s comparable to Gallian, with a little more flexibility in the topic sequencing and a greater, more integrated treatment of applications to coding theory and cryptography. (This last was something I was really looking for.) There’s even a free companion to the book which incorporates Sage, which I am sorely tempted to use as well because learning Sage has been a pet project of mine.

But what’s really different about this book is that it’s free, licensed under the GNU Free Documentation License. I am having the bookstore prepare print copies for the students — I asked the students if they wanted a print version in addition to the free PDF’s online, and they said “yes” — which the bookstore will sell for a whopping $16.95, just enough to cover the costs of copying and 3-hole punching the 400+ pages of the book. I’m happy because I found a book that really fits my needs; the students are happy because they get a good book too, for a tremendous bang-to-buck ratio.

In the long and contentious comment thread for my post about James Stewart’s new $24M mansion, I suggested that Stewart should consider topping off his impressive (and apparently lucrative) teaching and writing career by making his Calculus book freely available online for anybody who wants it. That suggestion was met with shocked incredulity: “If you had any idea how much work it was to write and maintain a textbook, you’d never consider making it free.” Well, I’m happy to report that hard work and good writing need not necessarily be mutually exclusive with giving it away.

In fact, as more well-written textbooks appear for free online — and there were even more free abstract algebra e-books I did not end up selecting — the commercial market might find itself in trouble.

* Actually, I requested the Hungerford algebra book, complete with a crystal-clear note that I needed to have it in hand by April 10 in order to be able to adopt it in time for our bookstore. To this date I have not received it. Another problem with commercial textbooks: the distribution model for review copies is dreadful. I’m always receiving multiple copies of books I neither need nor am interested in, and not getting the books I do need and am interested in.

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Filed under Abstract algebra, Calculus, Education, Teaching, Textbook-free, Textbooks

Questions about the algebra course

Jackie asked a series of good questions about the textbook-free modern algebra course and some of the student outcomes I was seeing in it. I tried to respond to those in the comments, but things started to get lengthy, so instead I will get to them here.

Do you think the results are only a result of a textbook free course?

To repeat what I said in the comments: I think the positives in the course come not so much from the fact that we didn’t have a textbook, but more from the fact that the course was oriented toward solving problems rather than covering material. There was a small core of material that we had to cover, since the seniors were getting tested on it, but mostly we spent our time in class presenting, dissecting, and discussing problems. We didn’t cover as much as I would have liked, but this is a price I decided to pay at the outset.

Most traditional textbooks don’t lend themselves well to this kind of class design. The ratio of text to problems in a typical textbook is probably something like 5:1 — a lot higher than that in some books. When you have a book in the course, it almost forces itself into the center of the class universe and everything tends to revolve around it, and take on its flavor. When the book spends most, almost all, of its pages on stuff for students to read rather than on problems for students to solve, then I guess it’s possible to have a problem-solving oriented class, but you’re going to be swimming upstream the whole way.

It works better, I think, to have no central book — and instead, provide problems via the course notes with just enough information to solve the problems. And if the students need more information, make it an assignment for library research or web queries.

Were there any negative outcomes? Anything you didn’t like as a result of choosing to structure the course in this manner?

There are some important algebra topics, in rings and particularly in fields, that are not going to get the time they really deserve. And I had to cut short or cut out some topics in group theory that are normally standard fare. At least, I see this as a negative; whether it really makes a difference in the long run is yet to be determined.

The way I select students to do course tasks in class basically involves randomly ordering the students and having them attempt the problems one after the other. It seemed like several times, students who had not presented much ended up first on the list on the days they didn’t have something and last on the list on the days they did. Call it bad luck or Murphy’s Law or what-have-you; but I didn’t like how there was no mechanism for making sure the lower-scoring students got more chances to work.

Some students in the class still struggle with basic problem-solving skills and writing proofs. I think they have enough education to carry out successful problem-solving on proofs most of the time. But not having me lecture has meant that they don’t get to see professionally put-together proofs very often unless they go do some reading.

And I think that this course structure caused stress and even ill will among the students who were not used to having so much personal responsibility in their college work. I think that’s an unintended consequence of implementing a course design that is basically sound; I regret that it happened, and I’d like students to have a more uniformly positive experience in the class, but I’m not going to change the basic course design.

Would you do this again?

You bet, although I believe this way of running the class works in some situations and wouldn’t work in others. I thought about running my differential equations class next semester like this, but that course is so focused on methods that a blind application of this course structure onto that course doesn’t seem appropriate. Maybe I’ll come up with some variant that works.

What would you keep the same? What would you change?

I would definitely keep my method for assigning problems to students, my rubric for grading course tasks, and just the overall procedure for running the class sessions that I used. And I’d keep the feature where students get to choose the weights on the various assessments.

I’d do a little more with the course wiki. Right now students are expected to write up their solutions to course note tasks on the wiki, but there is no point value in doing so nor a penalty for not doing so. The exams are open-wiki, though, so there is some incentive for writing results up well. But I think I would make the posting of solutions mandatory and enforce the rule.

I’d also try to have a complete set of notes before the course began. I have been writing things as I go, and it’s led to some snafus I could have avoided.

I might try writing the course notes so that rings and fields come first.

I’d seriously consider having proof techniques be offered as the subject of weekly help sessions or additional course work. Some students are still struggling with basic problem-solving techniques, and they really need more help than what they are asking for.

That’s that for the questions. Any more?

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