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March 30, 2014

Fourier Series and Flipped Classrooms

Posted by Tom Leinster

Term is nearly over, which for me means the end of the 4th year Fourier Analysis course I’ve been teaching for the last couple of years.

I was fortunate enough to take over the course from Jim Wright, a genuine expert on the subject, and I inherited a great set of notes from him. But I felt the need to make the course my own, so I’ve been writing my own notes, which I’ve just finished: notes here, plus accompanying problem sheets. They’re mostly about convergence of Fourier series, with a delicious dessert of Fourier analysis on finite abelian groups.

But what I wanted to write about here — and get your opinions on — was not Fourier analysis, but some questions of teaching. This year, I’ve been (in the jargon) “flipping the classroom”, or at least partially flipping it (which reminds me of that mysterious substance, partially inverted sugar syrup, that you sometimes see on ingredients lists). I’d like to hear about other people’s similar experiences.

The allotted class time is two 50-minute “lectures” a week, plus one “workshop” (Edinburgh lingo for tutorial or exercise class) a fortnight. Here’s the routine for lectures:

  • At least a week before a given lecture, I make Latexed notes for that lecture available online.

  • The students are committed to spending at least an hour before each lecture reading over the notes.

  • I spend the first half of each lecture giving an overview of that portion of the notes, on the firm assumption that the students have done the prescribed amount of reading beforehand. Thus, I can spend time talking about the big picture rather than the mechanics, and I can concentrate on parts that seem likely to cause most difficulty rather than having to go through everything.

  • The second half of the lecture is interactive. We do different activities each time, e.g.:

    • solving exercises (individually, in small groups, or all together)
    • question and answer sessions
    • definitions quizzes
    • working on mathematical writing
    • identifying hard parts of the course and having students explain them to each other.

In a completely flipped classroom, all the time would be taken up with interactive work. The first half of each class isn’t too far from a traditional lecture, except for the data-projected notes and the prior reading by the students. That’s why I said it’s only partially flipped.

It’s the first time I’ve done things quite like this, so this year has been pretty experimental. In particular, before every lecture, I’ve had to come up with an idea for the second half of the class, and evidently some have been more successful than others. I’m also running out of ideas! It’s a good thing it’s the end of term.

Have you ever taught in a similar way? If so, what worked well, and what didn’t? Can you share some ideas for classroom activities?

Posted at March 30, 2014 1:32 PM UTC

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Re: Fourier Series and Flipped Classrooms

I’ve never done this, but I’ve been thinking for a while about trying it, so I’m interested to hear your experience! The main thing holding me back is a worry that the students won’t do the reading. Has that been a problem for you?

As for activities, are you trying to think of something new and different every week? I wouldn’t worry about repeating myself; in fact I’d be inclined to stick with only a few activities that work (once I’d found out which those were). Can you share your experience of which activities worked well and which didn’t?

The main new thing I’ve been doing this year in the classroom is “voting questions”(aka “clickers”, although we use a free smartphone app rather than actual clickers). I’ve been very happy with this overall, although it’s reinforced my fears that students won’t do assigned reading: even the easiest of voting questions based on the reading don’t get very good responses.

Posted by: Mike Shulman on March 30, 2014 3:58 PM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

Getting students to do assigned pre-class work is important in any setting, especially so in a flipped setting. I’ve found that the key is to make the pre-class work highly structured, with explicit and manageable learning objectives, high-quality resources to help them attain the learning objectives, and accessible exercises that help them see how close they are to attaining those objectives.

Just giving them a section of text and telling them to read it is likely to result in nothing, because students are novices and just don’t have the know-how to read a text. A strong role of the pre-class activity is to train students how to read that text.

I mentioned my blog in my other comment, but this post in particular addresses this issue: http://chronicle.com/blognetwork/castingoutnines/2014/03/04/the-inverted-calculus-course-using-guided-practice-to-build-self-regulation/

Posted by: Robert Talbert on March 31, 2014 12:57 AM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

… the key is to make the pre-class work highly structured, with explicit and manageable learning objectives, high-quality resources to help them attain the learning objectives, and accessible exercises…

I don’t have any trouble believing that. I must say that another reason for my reluctance to depend on pre-class work in a calculus class is my low opinion of pretty much all the calculus textbooks I’ve used. But I’m glad to see that you’ve made the resources from your course available online, so that those of us without the time to create such high-quality materials ourselves can try them out.

Posted by: Mike Shulman on April 1, 2014 5:27 AM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

Mike wrote:

The main thing holding me back is a worry that the students won’t do the reading. Has that been a problem for you?

A few weeks into the course, I conducted an anonymous survey on this and a couple of other things. It told me that some people were doing the reading and some weren’t; the sample size was too small to say more.

However, I’ve acted as if the students have done the reading, when I’m doing the overview in the first half of each lecture. I’m sure they know I know that they haven’t all done so (and maybe some of them will read these words), but I hope that continuing to act on that assumption encourages and rewards good behaviour.

Robert wrote:

Just giving them a section of text and telling them to read it is likely to result in nothing, because students are novices

These are 4th-year undergraduates, so they’re not exactly novices. Moreover, they’re a fairly good group, partly because I made it clear that in order to take the course, they needed to be on top of much of the material in their main 3rd-year analysis course. They’re certainly capable of reading the notes I gave them. And it hasn’t resulted in nothing: some of the questions they ask in class are obviously informed by the preparation they’ve done, and some of the anonymous feedback included comments to the effect that they found it useful to read the notes in advance.

That said, I’m very much aware that the approach would need to be decidedly different for, say, a large first-year class.

Mike wrote:

As for activities, are you trying to think of something new and different every week? I wouldn’t worry about repeating myself; in fact I’d be inclined to stick with only a few activities that work (once I’d found out which those were). Can you share your experience of which activities worked well and which didn’t?

Well, at the beginning I was bursting with ideas and tried out something new pretty much every lecture. By the second half of the course, I was taking my cue more from the course material.

As for what worked well and what didn’t, one lesson I learned was about Q&A sessions. The first time I tried it, I told them at the beginning of the lecture that during the second half, I’d answer their questions on the course so far. This didn’t work so well, as I hadn’t given them time to think about what questions to ask. (Obvious, really.) The second time, I gave them over a week’s notice and asked them to email questions to me in advance. This worked much better, I think.

I got anonymous student feedback via a couple of routes. Some of them said that they liked doing problems in class (so I did more of that). Some of them also said that they didn’t like being asked to reflect too much on the structure of the course, or on what they didn’t know. This was presumably provoked by exercises in which I’d asked them to draw diagrams showing how all the theorems fitted together logically, or identify areas of the course where they’re fuzzy.

This is one of those occasions on which I might believe that I know better than the students. Most of them hadn’t previously taken many courses in which there are such long chains of theorems depending on earlier theorems, so I wanted them to think carefully about how everything hangs together. To borrow Robert’s phrase, this is part of the know-how of reading a (relatively advanced) text.

Posted by: Tom Leinster on March 31, 2014 2:53 AM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

I’ve been using the flipped classroom for about five years now, in subjects ranging from calculus to CS courses. I blog about some of those experiences over at Casting Out Nines, which is hosted by the Chronicle of Higher Education (it should be linked to my name above).

The more I’ve refined the process, the better it’s worked (no big surprise there) but the most gratifying thing I’ve found is that my students are more engaged, better able to learn things on their own, and are becoming very enthusiastic about this approach to college work. They are coming to the conclusion, which I think is correct, that in order to really learn something, you have to struggle with it and experience it for yourself, rather than just sit in a lecture. (Although lecture has an important place at the table and always will.) I give about 4-6 workshops a year on the flipped classroom at various college campuses, and the more I work with faculty on this technique the more exciting it becomes. I really think in five year, we won’t be talking about “the flipped classroom” but rather just “the classroom”, as the flipped structure becomes normative.

I invite everyone to come on over to Casting Out Nines – I’m wrapping up a series where I debrief the completely flipped calculus 1 class that a colleague of mine and I did last year.

Posted by: Robert Talbert on March 31, 2014 12:50 AM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

Interesting. I enjoyed the post about the first few minutes of a class.

I’m already finding the phrase “flipped classroom” slightly annoying, even though I just used it myself. There’s something smug about it, like “I’m so original, I do everything differently”. (That’s not directed at you, of course, Robert!) So I too look forward to the day when we don’t say it any more.

Posted by: Tom Leinster on March 31, 2014 11:15 AM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

The common nomenclature among higher ed people is the “inverted” classroom rather than “flipped” – that term “flipped” somehow is more commonly used by K-12 teachers. I like “inverted” better because it describes the practice without being cute, but it doesn’t have the brand recognition that “flipped” does. But yeah, pretty soon we will all just be talking about “effective teaching” rather than “flipped” versus “unflipped”.

Posted by: Robert Talbert on April 1, 2014 1:25 AM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

K-12 teachers

= school teachers (up to age 18ish), to save anyone else looking it up.

Posted by: Tom Leinster on April 1, 2014 1:35 AM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

My only contact is with higher ed people, and I’ve only ever heard “flipped”, never “inverted”.

Posted by: Mike Shulman on April 1, 2014 5:07 AM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

I tried a variant of that partially flipped format in some Operational Research lectures some years ago. (The idea worked because of the type of subject and might not be adaptable to other topics.) I set questions on problem sheets (before they had met the method in lectures) that required them to try out some ideas as to how one might attack the (`real life’) problem, e.g. a transportation problem. Usually one can get a certain way in these problems using basic common sense and some earlier material (e.g. linear progamming). The problems are often combinatorics at least to get initial solutions, then iteration of some process gets the optimal solution.

The format I used would typically sketch out some ideas for them to try. It seemed to work well. It meant that when I talked about the algorithm later on in the lectures the students seemed to appreciate the way that the taught algorithm got around some of the difficulties that they had encountered when attacking things in the problem class. Once or twice students came up with really good points about why a particular method worked, insights that I had not met before. The lectues felt a bit more like a shared learning experience and they seemed to like that. I also wanted to emphasise that when you go to a maths class you should not leave your common sense outside!

Posted by: Tim Porter on March 31, 2014 7:06 AM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

The one thing I’m missing from your description above is a way for the students to assess their learning as they go through the material before class.

That is, either questions for them to answer as they go through the material, or (easy!) opportunities for them to ask questions or just say “I didn’t get this!”. This should be done in such a way that you get the chance to see their responses well in advance of the lecture so that you can take them into account during the lecture.

Then there is a definite incentive for the students to go through the material beforehand: if they do so then their questions will be answered in the lecture time. If they don’t, then they have to endure other people’s questions being answered without necessarily knowing all the background.

The problem I see with how you’ve done it is that half lecture at the start. I’m not sure that it serves any purpose. Either you assume that they’ve all read the material, in which case your best choice of activity is to find out what they didn’t understand and go directly for the jugular, so to speak. Or you assume that they haven’t, in which case those that have wonder why they bothered.

But it can still be useful to have a time when the expert speaks. So I would have a half lecture before they read the material giving them an overview of what’s coming up. Then they read the material and ask questions/indicate stuff they don’t understand. Finally, you take that up in the flipped part of the class.

Posted by: Andrew Stacey on March 31, 2014 2:56 PM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

Those are all interesting and provocative points. I’m a novice at this way of doing things, so it’s good to have this provocation.

One feature of this particular group is that they’ve been pretty reluctant to ask questions, despite the smallness of the group and (what I perceive to be) a nice friendly atmosphere. So much as I’d love the whole class to be as question-based as you seem to be used to, it definitely wasn’t happening this year.

At today’s lecture, I invited the students to leave comments here, so maybe, just maybe, one of them will come along and give some insight into this.

As to whether it’s best to give a summary mini-lecture before or after they’ve read the material, I actually gave them the choice at the start of term. But I did it after expressing my strong belief that reading before the lecture was the best way to go, and they voted accordingly. My principal thoughts here were that (1) the more you’ve struggled with material by yourself, the more you’ll get out of hearing someone speak about it in big-picture terms, and (2) if they haven’t read the notes before I do the overview, I’ll have to go over the definitions in at least rough terms, whereas if they have read the notes beforehand, that frees me up to do the kind of broad brush-stroke speaking that I think lectures are well-suited to.

Posted by: Tom Leinster on March 31, 2014 5:15 PM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

In Sheffield we have started teaching our maths courses for first year engineers in this fashion. We seem to be calling it ‘blended learning’ rather than ‘flipped classrooms’, I forget why.

We realised there was a need to change the way we were teaching the engineers as the traditional approach did not seem to be inspiring the students. We also have a massive spread of ability in these courses: some of the engineering students are at least as good at maths as our best maths students and some of the engineering students are quite a bit weaker at maths. This resulted in more extremes of feedback of the form “the lectures go way too slow” and “the lectures go way too fast”.

The traditional format was two hours of lectures and one hour of problem class, where the students sat and work on the standard problem sheets, putting up their hand to ask for help when they got stuck. Attendance at the problem classes tended to drop off rather quickly, leaving an enthusiastic core, but indicating that many were not getting much out of these classes.

We decided to switch to a mix of videos and two problem classes (of a more interactive nature) per week. Based on the local Catster knowledge, we opted for ten minute videos. The students have a couple of days before the problem class in which they must watch three ten-minute videos.
A group of about seven of us recorded the videos between us. Generally the videos are in the Catster format of one person talking in front of the board. Some people edited in some computer graphics, but in general there was minimal editing.

At the end of each video the students must answer three online questions. These are generally quite simple questions that test mainly that the students have watched the video. Our online testing system is built on maple, so the questions can be appropriately randomized and require mathematical input, as opposed to just multiple choice. As an encouragement to do the questions, the marks for them count for a small amount of the final grade.

The problem classes begin with the person leading the class giving a five-minute summary of the videos. They then go through a couple of simple questions with input from the class. The students then get a sheet of three-ish questions based on the videos they’ve watched, but quite varied in scope: from a little bit of basic material to some more interesting and challenging questions. A lot of effort went into trying to think up good, balanced, thought-provoking questions. The students can work on these in groups and in any order. This leads to more cohesion between the students, given that they are working on roughly the same problems, rather than some racing ahead. In the old system the problem classes included a lot of drill problems which the students are generally better working on at home. Indeed they still have the standard problems which they are encouraged to work on at home.

Unfortunately, because I’ve been teaching other courses I haven’t lead any of the problem classes, but I hear that most students seem to be enjoying them and initial feedback seems to be very good indeed. We’ve been testing this on a relativity small cohort of engineers this year, but intend to expand out to most of the first year engineers next year.

Posted by: Simon Willerton on March 31, 2014 6:46 PM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

Wow, that’s really impressive! It sounds great, and I’m not surprised you’ve had such good feeback.

It also sounds like a lot of work for you and your colleagues. No one’s really saying it here, but one of the disadvantages of all these creative teaching ideas is that they really take up time! Of course, I’m not saying it’s time wasted — if I thought that, I wouldn’t have done what I did with my own class.

Posted by: Tom Leinster on April 1, 2014 1:29 PM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

I’m part of a project at my university with the general theme of “Innovative Teaching”. One thing we’re investigating is the possibility of using more online resources.

I’m particularly interested in what web software people use for setting out these resources. What’re you (Simon) using? What got my attention was the video and maple integration[1].

(I also asked this on the new Mathematical Educators StackExchange site.)

[1] That’s the non-mathematical sense of the word.

Posted by: Andrew Stacey on April 7, 2014 7:22 PM | Permalink | Reply to this

Re: Fourier Series and Flipped Classrooms

We use AiM (Assessment in Mathematics) that Neil Strickland, who’s here in Sheffield, has done a lot to develop. It doesn’t use Maple TA, but it has Maple as the back end. I believe that there are maple worksheets that output the html. There’s a scripting language for authoring questions that uses a mix of Maple and LaTeX.

The documentation and the teacher’s interface on AiM are not great, so there is, as they say, a steep learning curve, but it is quite a powerful system, and it looks fine from the student’s perspective. I’ve used it a reasonable amount.

Neil has customized AiM so that videos can be embedded into quizzes.

I believe that Chris Sangwin forked off AiM to produce STACK which is based on Maxima, the point being, I think, that Maxima is open source whereas Maple is proprietary.

Posted by: Simon Willerton on April 7, 2014 9:57 PM | Permalink | Reply to this

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