Category Archives: Linear algebra

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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Speaking of the inverted classroom

On Wednesday, I gave a talk at Indiana University – Purdue Universty – Indianapolis (IUPUI, for short) to the teaching seminar for math graduate students on the inverted classroom. It was sort of a generalization of the talk I gave on the inverted linear algebra classroom back at the Joint Mathematics Meetings in January. Carl Cowen was in attendance at that talk and invited me to make the 20-minute drive from my house to IUPUI to do something like it, and I was happy to oblige.

Since putting the talk up on Slideshare yesterday morning, it’s gotten over 200 views, 2 favorites, a handful of retweets/Facebook likes, and is currently being highlighted on Slideshare’s Education page. So I thought I would share it here as well. Enjoy and ask questions!

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An M-file to generate easy-to-row-reduce matrices

In my Linear Algebra class we use a lot of MATLAB — including on our timed tests and all throughout our class meetings. I want to stress to students that using professional-grade technological tools is an essential part of learning a subject whose real-life applications closely involve the use of those tools. However, there are a few essential calculations in linear algebra, the understanding of which benefits from doing by hand. One of those calculations is row-reduction. Nobody does this by hand; but doing it by hand is useful for understanding elementary row operations and for getting a feel for the numerical processes that are going on under the hood. And it helps with understanding later concepts, notably that of the LU factorization of a matrix.

I have students take a mastery exam where they have to reduce a 3×5 or 4×6 matrix to reduced echelon form by hand. They are not allowed any technology on that exam. I’ve learned that making up good matrices for this exam is surprisingly tricky. My first attempt at writing the exam resulted in a nice-looking matrix whose reduced echelon form had mind-bendingly big fractions in it. I want the exam to be about row reduction and not fraction arithmetic, so I  sat down this morning and wrote this MATLAB function called easyRR.m which automatically spits out m \times n random integer matrices whose row-reduction process might involve fractions but which aren’t horrendous:

%% Function to create an mxn matrix that is easy to row-reduce by hand.
% Basic idea: Construct this matrix by building an LU factorization for it
% where both L and U have small integer values.
% R. Talbert, Feb 15, 2011

function A = easyRR(m,n)

%% Create the L in the LU factorization. This matrix encodes the elementary
%% row operations needed to get A to echelon form.

% Start with a random integer square matrix:
L = randi([-10, 10], [m,m]);

% Replace diagonal elements with 1's:
for i=1:m
L(i,i) = 1;
end

% Zero out all entries above the diagonal:
L = tril(L);

%% Now create the U in the LU factorization, using smaller integers so that
%% the back substitution phase isn't too bad.

% This creates an mxn random integer matrix and zeros out all entries below
% the diagonal.
U = triu(randi([-5,5], [m,n]));

%% The easy-to-reduce matrix is the product of L and U.
A = L*U;

Here’s a screenshot:

The fractions involved here have denominators no larger than 25, which is way more doable for students than what I had been having them work with (sorry, guys).

And, if you happen to have the Symbolic Toolbox for MATLAB, you can add the line latex(sym(A)) to the end and the function will spit out the \LaTeX code for that matrix, for easy copy/paste into the exam.

Anyway, I thought this was useful and so I’m giving it away!

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The inverted classroom and student self-image

picture of an e-learning classroom

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This week I’ve been immersed in the inverted classroom idea. First, I gave this talk about an inverted linear algebra classroom at the Joint Meetings in New Orleans and had a number of really good conversations afterwards about it. Then, this really nice writeup of an interview I gave for MIT News came out, highlighting the relationship between my MATLAB course and the MIT OpenCourseware Project. And this week, I’ve been planning out the second iteration of that MATLAB course that’s starting in a few weeks, hopefully with the benefit of a year’s worth of experience and reflection on using the inverted classroom to teach technical computing to novices.

One thing that I didn’t talk much about at the Joint Meetings or in the MIT interview was perhaps the most prominent thing about using the inverted classroom model on a day-to-day basis: how students react to it and change as a result of it. I was actually quite surprised that nobody at my Joint Meetings talk asked me a question about this, because honestly, the inverted classroom sounds great on paper, but when you start to imagine the average college student walking in on the first day of class and having this method of instruction described to him, it becomes clear that a significant amount of work is going to have to be done in order to get students — who are already resistant to any change from their accustomed modes of instruction — on board with the plan.

Students do tend to resist the inverted classroom at first. Some forms of resistance are more benign than others. On the benign end of the spectrum there are students with little experience with the course material or its prerequisites who get bogged down on the basic podcast viewing (which takes the place of in-class lectures in this model) or the accompanying guided practice, and instead of actively seeking a resolution to their question will wait for the instructor to clear it up — in class. On the other end is the student who simply doesn’t believe I’m serious when I say there won’t be any lecturing, who then doesn’t do the work, assuming I’ll bail him out somehow — in class. But in the inverted model, students are held responsible for acquiring basic competencies before class so that the hard stuff — what we refer to as assimilation — is the primary focus of the class time.

I break this distinction down for students, but not everybody buys into it. Those who don’t will have to undergo a learning process that usually looks like shock — shock that I won’t reteach them the material they were supposed to have viewed and worked on, while the lab assignment based on that material is going on. This can get very ugly in ways I probably don’t need to describe. Let’s just say that you had better not use the inverted classroom model if you aren’t prepared to put out a constant P.R. effort to convince students of the positive benefits of the model and constantly to assuage student concerns.

I’ve often wondered why students sometimes react so negatively to the inverted classroom model. I’ve come to believe it’s the result of a invasive, false belief that can arise in students about their ability to learn things independently of others — namely, that they simply cannot do so. I have had students tell me this to my face — “I can’t learn [insert topic] unless you lecture to me about it in class first.” Clearly this is not true. Toddlers learn their native language without formal instruction, just by assimilating (there’s that word again) the language going on naturally in their background. We all learn things every day without sitting in a classroom; we may seek out training data first through printed instructions, worked-out examples, YouTube videos, etc., but it’s almost never in a classroom setting. Learning new things on our own initiative and without formal instruction in a classroom setting is as natural to humans as breathing. Indeed you could say that it’s the capacity to learn in this way that makes us human. But somehow many students think otherwise.

Where does this belief come from? I think that it comes from its own instance of assimilation, namely the assimilation of a culture of programmed classroom instruction that takes place from roughly the first grade through the twelfth grade in this country. Students have so few experiences where they pursue and construct their own knowledge that they simply come to believe that they are incapable of doing so. And this belief is propagated most rapidly in mathematics. I’ve been reading in Seymour Papert‘s book Mindstorms: Children, Computers, and Powerful Ideas, and this quote hits this issue right on the head:

Difficulty with school math is often the first step of an invasive intellectual process that leads us all to define ourselves as bundles of aptitudes and ineptitudes, as being “mathematical” or “not mathematical”, “artistic” or “not artistic”, “musical” or “not musical”, “profound” or “superficial”, “intelligent” or “dumb”. Thus deficiency becomes identity and learning is transformed from the early child’s free exploration of the world to a chore beset by insecurities and self-imposed restrictions.

That last sentence (emphasis added) sums it up, doesn’t it? Deficiency becomes identity. Eventually, if a student is robbed of experiences of self-motivated learning, the student eventually adopts a self-image in which she is incapable of self-motivated learning. It is a false self-image that is ultimately dehumanizing.

Which is why I put such stock in the inverted classroom model. I think this method of teaching, along with other learner-centered modes of instruction like problem-based learning, is on the front lines in reversing students’ negative ways of thinking about how they learn. Students may (will?) chafe at the inversion at first. But in the MATLAB course at least, something really cool happened at the end of the semester. I made up a slideshow for students called “Five myths about how you think you learn that CMP 150 has busted”. Among the myths were “I can’t learn unless a professor lectures to me” and “I can’t learn on my own initiative”, and I gave concrete examples of work that the students had done in the class that contradicted these messages. In the end I showed them that through this inverted classroom process they had taken majors strides toward being confident, independent, skill learners and problem-solvers rather than just people who can play the classroom game well. And even the most skeptical students were nodding in agreement. And I think that makes it all worthwhile for everyone.

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Bound for New Orleans

Happy New Year, everyone. The blogging was light due to a nice holiday break with the family. Now we’re all back home… and I’m taking off again. This time, I’m headed to the Joint Mathematics Meetings in New Orleans from January 5 through January 8. I tend to do more with my Twitter account during conferences than I do with the blog, but hopefully I can give you some reporting along with some of the processing I usually do following good conference talks (and even some of the bad ones).

I’m giving two talks while in New Orleans:

  • On Thursday at 3:55, I’m speaking on “A Brief Fly-Through of Cryptology for First-Semester Students using Active Learning and Common Technology” in the MAA Session on Cryptology for Undergraduates. That’s in the Great Ballroom E, 5th Floor Sheraton in case you’re there and want to come. This talk is about a 5-day minicourse I do as a guest lecturer in our Introduction to the Mathematical Sciences activity course for freshmen.
  • On Friday at 11:20, I’m giving a talk called “Inverting the Linear Algebra Classroom” in the MAA Session on Innovative and Effective Ways to Teach Linear Algebra. Thats in Rhythms I, 2nd floor Sheraton. This talk is an outgrowth of this blog post I did back in the spring following the first non-MATLAB attempt at the inverted classroom approach I did and will touch on the inverted classroom model in general and how it can play out in Linear Algebra in particular.

Both sessions I’m speaking in are loaded with what look to be excellent talks, so I’m excited about participating. I’d be remiss if I didn’t mention that Gil Strang and David Lay are two of the organizers of the linear algebra setting, which is like a council of the linear algebra gods.

I’ll give Casting Out Nines readers a sneak peek at my two talks by telling you I’ve set up a web site that has the Prezis for both talks along with links to the materials I mention in the talks. And if you’re there in New Orleans, come by my talks if you have the slots free or just give me a ring on my Twitter and I’d love to meet up with you.

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Coming up in January

Fall Semester 2010 is in the books, and I’m heading into an extended holiday break with the family. Rather than not blog at all for the next couple of weeks, I’ll be posting (possibly auto-posting) some short items that take a look back at the semester just ended — it was a very eventful one from a teaching standpoint — and a look ahead and what’s coming up in 2011.

I’ll start with the look head to January 2011. We have a January term at my school, and thanks to my membership on the Promotion and Tenure Committee — which does all its review work during January — I’ve been exempt from teaching during Winter Term since 2006 when I was elected to the committee. This year I am on a subcommittee with only three files to review, so I have a relatively luxurious amount of time before Spring semester gets cranked up in February. A time, that is, which is immediately gobbled up by the following:

  • I’ll be at the Joint Mathematics Meetings in New Orleans from January 6–9. This will be my first trip to the Joint Meetings since 2002, and I’m pretty excited about it. I will be giving two talks, one in the MAA Session on Undergraduate Cryptology (PDF) about my five-day micro-unit on cryptology for freshmen and the other in the MAA Session on Innovative and Effective Ways to Teach Linear Algebra (PDF) on experimenting with the inverted classroom model in linear algebra. Both of those sessions are loaded with interesting-sounding talks, so I hope to attend the entire session. I also hope to catch up with friends I haven’t seen since, well, 2002 — and maybe connect with some new ones. If you’re attending, let me know!
  • The second iteration of the MATLAB course is coming up in the spring as well, and I will be doing some significant redesign work on it based on experiences and data from the first iteration. I’m constantly humbled and gratified by the interest and positive responses that the course has generated in the MATLAB community and elsewhere — and by how much interest and attention the course has received. I’ve had a chance to observe and talk to the alumni from the first run of the course during their Calculus III course that used MATLAB significantly, and their usage habits and feedback have given me some ideas for what should be positive changes in the course. I’ll elaborate on that later.
  • I am teaching Linear Algebra again in the spring, as I have done for the last 4-5 years, and this year I am targeting that course for a more robust implementation of inverted classroom techniques. A lot of the students in that course will be MATLAB course alumni, so they will be used to all that inversion. But I’ve had enough experience with peer instruction and classroom response system (“clicker”) use on the one hand from this past semester (which I never blogged about, and I’ll try to remedy that) and inverted classroom approaches in MATLAB on the other that Linear Algebra seems well-positioned to benefit from a combination of these approaches. I’ll be sketching out and planning the course in January.
  • Like I said, I used a lot of peer instruction and clickers in calculus this semester with great success (I think; at least the students say so). I’m teaching two more sections of calculus in the spring and will be refining my teaching using these tools. But calculus in the spring has a different flavor than calculus in the fall, so we will see how it goes.
  • What I’m reading this January: Teaching with Classroom Response Systems by Derek Bruff; Learning to Solve Problems by David Jonassen; The Craft of Research by Booth, Colomb, and Williams; and catching up on a mountain of articles that accumulated during the semester.
  • I’m also reading Geometry and Symmetry by Kinsey, Moore, and Prassidis leading up to an MAA review of the book. The “Prassidis” in the author list is Stratos Prassidis, who was my Ph.D. dissertation advisor.

Throw a couple of consulting projects on top of all that, and you’ve got yourself a busy January!

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Small proof of concept for inverted math classrooms

Geometric interpretation of the solution of a ...

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The very last topic in the linear algebra class this semester (just concluded) unexpectedly gave me a chance to test-drive the inverted classroom model in a mathematics course, with pretty interesting results. The topic was least squares solutions and applications to linear models. I like to introduce this topic without lecture, since it’s really just an application of what they’ve learned about inner products and orthogonality. Two days are set aside for this topic. In the first day, I gave this group activity:

The intent was to get this activity done in about 35 minutes and then talk about the normal equations — a much faster way of finding the least-squares solution than what this activity entails — afterwards. Then I meant to spend a second day on practice.

But, as they say, life got in the way — and we ended up spending the entire time just getting through the first 3/4 of the activity. We spent most of the second day debriefing the math in the activity, reducing my lecture on the normal equations to a 10-minute time slot… which I then screwed up hugely by making an intractable math error which I didn’t catch until time was up. To make things worse, this was the last day I had budgeted for covering material. So this was going to be on the final, but I had no time in which to teach it right.

Solution: Make a video containing the correct form of the lecture I was going to give, and a couple minutes shorter to boot:

…make another video with a more intricate example:

…and then give students some exercises to do and open the email/Skype/IM/office hours line for questions.

By the night before the final exam, the videos had about 15 hits each, some of which were from IP addresses my students would have. So they were being watched. On the final, I gave a mid-level exercise from the section on linear models in their textbook — an even-numbered one, so it was fairly unlikely they worked it.

The results? Out of 14 students taking the exam:

  • Two students had basically no progress on the problem — left it blank or just wrote down the problem.
  • Five students did NOT use the normal equations approach from the videos but the projection-oriented approach from the group work. Of those five, only two used the method correctly.
  • Seven students DID use the normal equations approach from the videos. Of those seven, six used the method correctly.

I’d like to conclude that the videos were more effective in conveying the subject matter than the group work was, and took less time. But of course there are confounding variables. The videos covered material that is less complicated than the group work covered. Perhaps only the most motivated students actually watched the videos, and their motivation rather than the video is what helped them to learn. But still, I think the inverted approach here had some kind of effect. Certainly the students who watched the videos were mostly (6 out of 7) able to pull off the method on the final, whereas less than half the students who relied upon what we did in class could do so with that other method.

Ideally, I’d put these videos out there, assign them along with some basic computation exercises, and then come in and do a group work assignment similar to what I used. With these kinds of results, I’m a little more emboldened to do so with some classes this fall. But more on that later.

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