## Using slides in math classroom

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I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this.

Obviously, slides make it possible to produce and show clear graphs/pictures (which will be particularly advantageous in the course I will teach) and doing all sorts of things with them; things that are difficult to carry out on a board. On the other hand, there seems to be something about writing slowly with a chalk on a board that makes it easier to follow and understand (I have to add the disclaimer that here I am just relying on only a few reports I have from students and colleagues).

It would very much like to hear about your experiences with using slides in the classroom, possible pitfalls that you may have noticed, and ways you have found to optimize this.

I am aware that the question may be a bit like the one with Japanese chalks, once discussed here, but I think as using slides in the classroom is becoming more and more common in many other fields, it may be helpful to probe the advantages and disadvantages of using them in math classrooms as well.

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Now you can delete your comments about making it community wiki! – Kevin O'Bryant Nov 4 2011 at 20:49

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I put a similar "math teaching style" question on math.stackexchange rather than the "research oriented" Mathoverflow, and it was closed as being too subjective and not "mathy" enough (despite getting many upvotes). Maybe I should move the discussion to MO ! – gordon-royle Nov 5 2011 at 23:00

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@Andrew - I won't... just amused by the erratic application of the "rules" about questions that are not research-level mathematics questions (e.g. "Most memorable title"). The fact that this question and similar soft-questions get many upvotes and participation from great mathematicians such as Terry Tao, might perhaps lead to some introspection about whether the rules are too rigid. But I'm not going to get evangelical about it. – gordon-royle Nov 6 2011 at 12:39

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There are no rules, only guidelines. I encourage you to join meta where you'll find that we debate them quite often (and whinge about exactly the point that you've just made). – Andrew Stacey Nov 6 2011 at 14:20

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I'll post this as a comment rather than an answer, but Joseph Gallian has some recommendations and a short list of pros and cons in his Advice on Giving a Good PowerPoint Presentation (d.umn.edu/~jgallian/goodPPtalk.pdf). Might be of interest. – J W Nov 9 2011 at 17:49
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## 17 Answers

I think you already touched on the two main points: pretty pictures are so much better than anything done on a chalkboard is the pro, but you cannot decently unwind any argument on slides.

I've used them intensively, I do it a lot less now. (Here's a con you did forget about: they take a lot of time to prepare, even when you're only revising them.) If the room lends itself well to it, the hybrid method is best: use the slides only when they beat the board. Rooms that have a screen in the corner, rather than in front of the board, are best for this.

Also, it seems that it's easier to fall asleep to slides than to a lecture, so be aware of that. Make sure that the room is never too dark (the quality of the screen material can be critical here too: good screens should be readable in full light). And switching your routine, never showing slides for too long, helps keeping the students awake.

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Yes, slides take a lot of time to prepare, and not all students realize or appreciate this. They may think the instructor used something ready-made, similar to commercially available lessons on slides at the grade or high school level (which they may have seen in use). I know someone who took great pains to prepare slides from scratch for a post-calculus course, only to get teaching evaluations like "He teaches from Power Point, give me a break". This is to emphasize the point you are making, not to discourage the OP from using slides. – Margaret Friedland Nov 4 2011 at 16:31

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I've also seen lecturers who "borrowed" slides from people who have taught the same course in the past. This inevitably leads to the embarrassing "what I think they're trying to say here is..." – Michael Lugo Nov 4 2011 at 18:36
Borrow slides by all means, but edit them yourself first! That takes time, but a lot less time (as I know from reusing my own slides and usually changing them a bit). – Toby Bartels Nov 7 2011 at 8:42

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Slides can, in principle, enhance a lecture, but there is one important difference between slides and blackboard that definitely needs to be kept in mind, and that is that slides are much more transient than a blackboard. Once one moves on from one slide to the next, the old slide is completely gone from view (unless one deliberately cycles back to it); and so if the student has not fully digested or at least copied down what was on that slide, he or she will have to somehow try to catch up in real time using the subsequent slides. Often, the net result is that the student will become more and more lost for the remainder of the lecture, or else is spending all of his or her time transcribing the slides instead of listening in real time.

In contrast, given enough blackboard space, the material from a previous blackboard tends to persist for several minutes after the point when one has moved onto another blackboard, which allows for a less frantic deployment of attention and concentration by the student.

If one distributes printed versions of the slides beforehand, then this difficulty is mostly eliminated. Though sometimes it takes a few lectures for the students to adapt to this. Once, in the first class in an undergraduate maths course, I said that I wanted my students to try to understand the lecture rather than simply copy it down, and to that end I distributed printed copies of the slides that I would be lecturing from. (The slides were in bullet point form, and I would expand upon them in speech and on the board.) I then found that for the first few lectures, the students, not knowing exactly what to do with their time now that they did not have to take as much notes, started highlighting all the bullet points on the printed notes. It was only after I threatened to distribute pre-highlighted lecture notes that they finally started listening to the lecture (and annotating the notes as necessary).

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I'm experiencing some of this right now. My students (at UCLA, by chance) actually told me that they want the slides to print before class, so they could annotate them. Others complained at the beginning that the slides vanish too quickly. After changing the format to make the lecture slower and more printable, I find that my classes have been much better. – Ryan Reich Nov 5 2011 at 23:44

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@Terry: I'm curious, did you end up liking the lecturing approach described in your last paragraph (did students understand more), or did you drop it for some reason? (From my personal experience as a student: when I don't take notes I feel like understanding more during the lecture but forgetting faster afterwards. When I take notes I can't follow that much but have the impression that the knowledge stays for longer. I wonder if that's just my impression or if there are studies proving this effect.) – Michael Nov 7 2011 at 15:18

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I haven't taught these sorts of classes for a while, but certainly if I had to teach a course for which I had previously already prepared such slides, I would try doing so again. It is true though that one needs to prevent material being forgotten after the lecture - good homework sets are one important way to combat this. – Terry Tao Nov 7 2011 at 23:52

I would never, never use slides for a course.

That said:
I do sometimes show my student pictures taken from the web.
For example, I recently showed this picture to the students in my group theory class in order to illustrate the isomorphism between $A_5$ and the group of symmetries of a dodecahedron.

Also, I sometimes prepare animations with Geogebra that I then show during class. Here's an example (click and drag the blue node). Of course, it's even better to create the graph in front of the students: Geogebra is good for that. My philosophy is that students should be shown things being created, not ready made. But I'll admit that this is not always possible...

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I made a model in paper of the five tetrahedra (following web.eecs.utk.edu/~plank/plank/origami) Nothing beats actually rotating the thing with your hands :D – Mariano Suárez-Alvarez Nov 5 2011 at 2:53
Anyhow, +1. ${}$ – Mariano Suárez-Alvarez Nov 5 2011 at 2:54

I'm going to try to answer the actual question rather than saying whether I think that chalk or projector is better. That "question" being:

It would very much like to hear about your experiences with using slides in the classroom, possible pitfalls that you may have notices, and ways you have found to optimize this.

(Though I'm curious about the request for ways found of optimising pitfalls!)

I switched to using beamer slides 2.5 years ago. I'm partway through the fifth course that I've given using slides (and the course immediately prior to those was given on chalkboard but having prepared them as slides - a half-and-half experiment). By-and-large, I would say that I give better lectures using the slides than I used to when giving chalk talks. The following is a fairly disorganised list of my thoughts on both why I chose to switch and things that I've learnt in the process. I hope that this will be of use to you. Feel free to contact me for more details, and we've also recently been discussing this a bit on the nForum.

1. A big reason for me switching was that I teach in English in a Norwegian University. Although the students have excellent English, it is not their native language. It takes them longer to copy from the board, and their error rate is higher, so more time in a chalk-talk is wasted waiting for them to catch up than I felt I could allow. Giving the lecture using slides meant that I had much more control over where the students were focussing at any particular time (mainly, I wanted this to be on me).

(To be clear: the time taken was in addition to the necessary time for students to process ideas that they've just been told about. Of course, pauses are necessary. But pauses by happenstance - because the students are busy copying the board - are not the best pauses.)

2. As a consequence, I always make my slide notes available beforehand. Admittedly, sometimes it was at 11pm before an 8am lecture, but no-one's perfect! They can get the actual presentation, a compressed version (the `trans` option), and a handout version (they are strongly encouraged only to print the latter). That way, they can read in advance what I'm going to show them, and they can bring the handout version along to add any additional notes if they wish.

3. The handouts are not a substitute for going to the lecture. The slides are not a summary of the lecture, they are what I want the students to be able to see while I am talking to them about something. Ideally, when the students look at the notes afterwards then they will be able to remember (more-or-less) what I've said. But if they weren't at the lecture then they won't have anything to remember so the handouts will be of less use (not of no use, it will still say what topics were covered so they can find out about them by other means).

4. Lectures never go completely as planned. But never use the chalk-board and the screen. Whenever I see someone doing this at a conference I want to run out of the lecture hall screaming. Not only will the lighting be completely different for both, but also the students will have the wrong mindset and will take time to make the switch. Use a system whereby you can write on the presentation (and can bring up blank pages if needed). You can even leave deliberate gaps if you want! As well as not requiring a change in gear, you can then make the annotations available afterwards (and have an easy record of the annotations that you made when you revise the slides for next year). I've used xournal (for Linux), jarnal (when forced to use Windows), and am currently using an iPad (despite what's said elsewhere, this is extremely usable for this). (Incidentally, I'd say that going the other way is acceptable: if you are primarily using the board and then want to show a couple of fancy pictures then so long as it doesn't take an age to set-up the projector then it's okay.)

5. Practise. Get a system so that your writing on the screen is acceptable (don't worry about perfect), you know how your program works, and you can change pages easily (preferably without looking at the machine).

6. Yes, it usually takes longer to prepare the slides - first time. But once you're used to the flow of writing a beamer presentation then that aspect doesn't actually add that much more. What probably adds the most time is that you are now forced to completely prepare the lecture in advance, rather than "winging it" and claiming that it is "good for the students to see the professor make mistakes". (You can probably guess my reaction to those!) It can take some effort to get a really nice system, I think I have one, and now it doesn't take me long to prepare a presentation.

On that note, preparing your notes in LaTeX makes it much easier to prepare it in "layers". First, lay out your lesson plan (you do have one, right?), then add the frame titles, finally add the content of each frame. Then go back and adjust the lesson plan according to what did and didn't fit as you expected. (And it's possible to produce the lesson plan from the same source as the presentation.)

And when you come to reuse the slides, it's much faster.

7. Think always "What can the students see right now?". If you want them to be able to refer to more than you can fit on a slide, consider giving them a "cheat sheet" handout as well. Slightly ironically, giving the lectures using LaTeX means that I am much more aware of how the presentation looks, something that is just as important as what is in it.

8. As hinted above, my slide notes would not form a good set of "traditional lecture notes" from which to revise. But then I don't believe that the chief aim of a lecture should be to produce that. Again, consider using other methods for this. For my course, I have a wiki where I can put more lengthy arguments. I use homework questions to "force" the students to read the wiki.

That's all that I can think of right now. You can get an idea of what my lectures look like by visiting the home page of my current course: http://mathsnotes.math.ntnu.no/mathsnotes/show/TMA4145+Home+Page.

What I've said above is phrased a bit like advice, but it's really just a list of things that spring to mind when I think about how I've adapted. I will give one genuine piece of advice: don't base your lectures on what worked best for you. The reason why should be obvious! But to illustrate the absurdity, let me note that the undergraduate course in which I learn the most and where I really feel that I understood and still understand the topic the best, was the worst lecture course that I ever went to. Why? Simple: because I couldn't learn from the lecturer, I was forced to go and learn it by myself. So now I mumble, write illegibly, stop halfway through a proof, and get wildly sidetracked by irrelevant questions - because that's what worked best for me!

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What iPad software do you use? Do you use a stylus - I can't write legibly with my fat fingertip.. – gordon-royle Nov 5 2011 at 22:56
I appreciate all these tips (advice or not). I'm just trying for the first time to give a beamer course and this is a very timely question for me. – Ryan Reich Nov 5 2011 at 23:39
Anyone curious as to my iPad set-up should take a look at this: tex.blogoverflow.com/2011/10/i-tex-therefore-ipad – Andrew Stacey Nov 6 2011 at 8:15
@Andrew: I am very amused by your intransigence. I can understand drawing personal battle lines and saying: "I won't do x", but your answer here sounds a lot more like: "no-one should do x". All rules have their exceptions, and I'm quite willing to let lecturers operate on their own terms. I must have seen every single approach to lectures done brilliantly, and every one of them done poorly, depending on who was giving the talk... – Thierry Zell Nov 7 2011 at 16:30
Thierry: Did you read the first sentence of my last paragraph? Part of my intransigence is the fact that I always feel in these discussions that there is a strong "Don't use slides" answer (even though the questioner asked directly for experiences of using slides) so I was partly trying to counter it. I wish I had seen every single approach to lectures done brilliantly (and poorly). I'm afraid that my experience has been very limited so I find myself having to invent my own techniques, and when I do I like to share what I've learnt. – Andrew Stacey Nov 7 2011 at 19:00
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I use a hybrid version for some of my classes which take place in a room that allows this: I use computer slides (and animations, computations, etc.) and the board. I learned this from my colleague Serkan Hosten, and it works really well in some classes. E.g., I use slides for definitions and theorems (including the relevant ones from the previous lecture) but then work out examples and proofs on the board. This has the obvious advantage of spending time on exactly the items that need time and just the right pauses to get digested, but it also has nice side effects: e.g., the statement of the theorem will stay on the screen even if I'll have to clean the board.

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+1: "the statement of the theorem will stay on the screen even if I'll have to clean the board". That's excellent practice. (Only very experienced teachers will know not to erase the board entirely, and keep the important definitions in some corner -- this requires a lot of skill). – André Henriques Nov 4 2011 at 20:21

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Only very experienced teachers will know not to erase the board entirely, and keep the important definitions in some corner -- this requires a lot of skill ... and a lot of room... – Thierry Zell Nov 4 2011 at 21:54

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And room is harder to get than skill! – Mariano Suárez-Alvarez Nov 5 2011 at 2:52

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@Thierry: I agree with you that it's easier to do this with a big blackboard that with a small one. But it's not impossible. The hardest thing is to arrange that, when you first write the definition, you do it in a corner (and without taking too much space). Only then will it be possible to erase everything but that definition, and go on with the lecture. – André Henriques Nov 5 2011 at 10:55

I just decided this quarter to use slides for my calculus class, a large-lecture course of the sort I'd never done before; I figured it would be easier to see the "board" if it were on the big screen. Here is the progression of my mistakes and corrections:

• My first lectures had too many words. Slides are great for presenting the wordy parts of math, because they take so long to write and then the students have to write them again. What is not great about them is how much they encourage this behavior.

• Since I was giving a "slide presentation" or a "lecture" rather than a "class", my mindset was different: sort of presentation-to-the-investors rather than gathering-the-children. My slides went by too quickly.

• I eventually slowed myself down by basing the lectures around computations rather than information. Beamer is pretty good (though not ideal) for this, because you can uncover each successive part of an equation. If you break down your slides like this, it is almost as natural as writing on the board.

• My students themselves actually brought up the point that Terry Tao mentioned in his answer: the slides were too transient. They also wanted printouts. Having to prepare the slides for being printed in "handout" mode changed how I organized them: for one, no computation should be longer than one frame (something I should have realized earlier). Also, there should be minimally complex animations, since you don't see them in the printout.

• Many of them expressed the following conservative principle: they had "always" had math taught on the board and preferred the old way. So I've started mixing the board with the slides: I write the statement of the problem on a slide, solve it on the board, and maybe summarize the solution on the slides. This works very well.

• Now I can reserve the slides for two things: blocks of text (problem statements, statements of the main topic of the lesson) and pictures. TikZ, of course, does better pictures than I do, especially when I lose my colored chalk.

Preparing these lectures used to take me forever. Using beamer does require that you learn how it wants you to use it: don't recompile compulsively, because each run takes a full minute, and don't do really tricky animations. Every picture takes an extra hour to prepare. If you stick to writing a fairly natural summary of a lesson, broken by lots of \pause's and the occasional `\begin{overprint}`...`\end{overprint}` for long bulleted lists, an hour lecture will take about two hours to prepare.

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Very nicely detailed answer! I like the slide summary of the solution, since it goes back to that cardinal principle of teaching: tell them what you're going to tell them, then tell them, then tell them what you just told them. On another note: Beamer is an incredibly flexible tool when it comes to handouts, and you can easily include slides that are handout-only if you need to convey extra information (e.g. a few frames from an animation.) Also, compiling in draft mode can really speed up those slides preparations! – Thierry Zell Nov 6 2011 at 1:51
I think I like your points in the order that you gave them. The first two are extremely important and ones that I should have thought to list as the major change from my notes from last year is that I'm going through taking out all unnecessary verbiage. From your description at the end, I'd say that you were now giving a board talk supplemented by slides. I'm surprised to hear that UCLA students are so conservative! I'd've thought they'd be more open to someone trying new ways of teaching them. – Andrew Stacey Nov 6 2011 at 8:25
@Andrew: Of course, any communications with students are filtered through the "boldness" mask, and you always hear from the discontents. I don't know what the silent majority thinks, and I can't really find out, though the other day in office hours, I had one guy who said he was surprised to see us "still" using blackboards in college. I asked what he expected, perhaps slide presentations? – Ryan Reich Nov 6 2011 at 15:40

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You can find out what the "silent majority" thinks. As Feynmen was once told: you just ask them. It's amazing how many ideas and techniques there are for getting information out of students. The problem is that no-one ever tells us about them! – Andrew Stacey Nov 6 2011 at 18:43

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Just ask! Half way through the lecture then stop, hand out a questionnaire, and say "I'd really appreciate it if you could all take a couple of minutes to fill this in. Although you all fill in the student evaluations at the end of the semester, that doesn't help a lot with this semester." Give them a candy bar while they do it. Go out of the room while they do it. Have them get into small groups to discuss what they do and don't like. Get someone else in to ask them the questions. Ultimately, if you show that you care and that you will take on board what they say, they'll say it. – Andrew Stacey Nov 7 2011 at 8:52
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Full disclosure: I stole the following idea from my wife.

For some courses, like calculus, I will create slides with beamer, leaving blank spots to fill in during class. I then print the slides out on paper and present them with the document camera during lecture. When I get to an example, I will work it out by writing on the paper during class, and have it projected in real time. This approach combines the advantages of blackboard talks where you work things out in realtime, with the advantages of beamer presentations where you can present nice graphics and also have an outline to limit getting distracted and wandering off on tangents.

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I get all my best teaching ideas from my wife too. – Andrew Stacey Nov 5 2011 at 22:19

I took a course at PCMI some years ago from David Perkinson (Reed College). He did an amazing job and single-handedly convinced me it was possible to teach well from slides. Check out this link to see examples of his slides. As the other answers have mentioned, it seems necessary to use slides only in conjunction with the board. Perkinson did this, but also included a useful trick: he created handouts from the slides for students to write on, but left blank spots in those handouts so they had to write the proofs themselves based on what he said, showed on the slides, and wrote on the board.

Professor Perkinson is also a wizard of sorts with mathematica, and he was able to create awesome graphics using it. I don't think his mathematica code is online, but I'll bet he'd be willing to share if someone emailed him. He may also have tricks to reduce prep time, as this was the sort of thing he liked thinking about.

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I really enjoy Mathematica's graphing capabilities. Even without being a wizard, Mathematica strikes me as especially good (better than Maple, I must admit) when it's time to combine different graphics into one picture (say curve plus scatterplot plus text). Especially useful when designing graphics for the classroom or a test. – Thierry Zell Nov 4 2011 at 16:47

I just finished teaching a course on linear algebra to non-math students. I used a combination of latex-beamer slides and blackboard. One advantage of the slides was being able to do examples of Gauss elimination and inversion of matrices quicker than on the blackboard and without making mistakes. On the other hand, I feel that slides can easily make a lecture less interactive.

And, I must agree with Thierry Zell: it took quite some time to prepare these slides, even though I could adapt the latex sources from the previous people teaching this course.

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My solution is to use a tablet PC (the pen-enabled kind, not the modern entertainment tablets like the Ipad),hooked up to a data projector.

I have "lecture templates" which contain the copying intensive stuff (statements of theorems, definitions, graphs, complex diagrams) on the page, along with plenty of blank space for annotation. Those are on a website prior to the lecture. The students print them off at home, and bring them to class. I then annotate the lecture notes (using a pdf annotator and the tablet pen) and the students take notes as they wish.

This, I feel, combines the benefits of having some complex material prepared ahead of time with the benefits of having arguments, calculations etc. developed in real time, rather than canned in advance. So it avoids the canned slides-whizzing-by problem.

The only disadvantages I can see are the limitations of screen size. Sometimes nothing replaces the virtue of a big whiteboard, and having every part of a long development in front of your eyes all at once. In that case, I use a whiteboard.

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An iPad works just fine, by the way. – Andrew Stacey Nov 5 2011 at 22:18
The quality of "inking" I've seen with the after-market pens for the Ipad is pretty miserable. Not surprising given that they weren't designed for digital inking (and given Job's antipathy to the digital pen.) But perhaps there's something better I haven't seen. – GMark Nov 6 2011 at 1:51
I don't know what "inking" refers to. Certainly I can get better control with a graphics tablet (no need for an actual tablet computer), but it's all a balance. I get good enough quality with an iPad that it's other benefits mean that on the whole it is better than my laptop. (See my comment on my post if you're interested in more details on my iPad experience.) – Andrew Stacey Nov 6 2011 at 8:20
I don't know what "inking" is either, but for the record, the quality (response, especially) of input devices for the ipad is, in my experience, extremely variable. My first cheap stylus didn't cut it for very long. – Thierry Zell Nov 9 2011 at 19:12

If you intend to post your slides online after class then you run the risk of students not even taking notes/digesting the material on their own (I've had this feeling myself) or feeling that they don't have to attend class. This is obviously a con but the other side is that the students then have a good outline of what you talked about in class with your emphasis included.

I second Thierry's comments.

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Frankly you suggesting to keep some secrets from the students to make them attend the class I think it is a wrong idea. – Anton Petrunin Nov 5 2011 at 1:20

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@Anton: I don't think what is suggested here is keeping secrets. But providing slides handouts is something that I've only ever done reluctantly, because the danger is to provide a fake sense of security to students, that the handouts contain everything they need to know. They seldom do. – Thierry Zell Nov 5 2011 at 2:36

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@Anton I am not suggesting keeping secrets from the students but rather not giving them a finished product along with the fake sense of security that Thierry has mentioned. My issue with handing over the finished notes/slides is that it does not provoke the students to spend time thinking instead of memorizing. – BSteinhurst Nov 5 2011 at 17:44

I have a story in the middle. I hurt my right shoulder over time, by 1994 it was simply too much to write on a blackboard, at least overhead. So, pre-Beamer, I wrote up these slides on transparencies with colored pens. These were unusually well-prepared lessons for me, I had everything worked out, it was all clearly my work, and I still had plenty of blank slides on which to write new material when needed. That is the hardest I have ever worked on course preparation.

They did have class questionnaires, sent to administration and never seen by me, later the chairman told me how very much the students hated the slides. They were never fond of me but I think that was a separate item, the slides made it worse than it would have been...I suppose my question now is, would things have been different if I also gave each student photocopies of the slides for that day?

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The secret is giving them icecream... – Mariano Suárez-Alvarez Nov 5 2011 at 2:49

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Given where I was working, chicken-fried steak. – Will Jagy Nov 5 2011 at 3:01
Chicken fried ice cream? – Spice the Bird Nov 6 2011 at 1:20
ice fried chicken cream? a bad joke perhaps. – Shripad Nov 6 2011 at 11:23

I've already given my opinion, and this is more of a remark: how the pros and cons are weighed between blackboard and slides should be influenced by a whole collection of classroom factors, and the first one among them should probably be class size.

This is a rather obvious remark, but I thought it was worth pointing out; Jaap Eldering's answer brought it to the forefront for me, because he mentioned doing examples on slides to avoid making mistakes, and my first reaction was: "making mistakes in class is good!".

And then it occurred to me that I can use mistakes in the classroom fairly effectively because I only teach small classes. In a big classroom, I would simply not be able to receive instant feedback efficiently enough to do this as well, and I would not be comfortable trying.

In a very large lecture hall, the blackboard will often lose a lot of its advantages given how large you have to write.

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Indeed there's didactical value in making mistakes. But e.g. in my case of a simple calculation error when inverting a matrix, I think the time consumed to find the error outweighs to advantages. – Jaap Eldering Nov 5 2011 at 10:13
@Jaap: of course, especially since finding a mistake in a matrix computation can take such a long time. – Thierry Zell Nov 5 2011 at 14:14

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I disagree that there is a didactical value in making mistakes in a lecture. There is a didactical value in showing how one might go about solving a problem including all the mistakes that you might have made or that you did make when you first tried to solve the problem by yourself. – Andrew Stacey Nov 6 2011 at 18:45

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@Andrew: you're very free to disagree, of course. But I learned a tremendous amount on how to fix algebraic mistakes from a fantastic Physics teacher who could never write two lines without dropping a sign or losing an exponent, yet could go back and (1) explain how he could tell there was a mistake in the result and (2) track down every single one, and thus accurately derive any physical law you care to name. He did not do it deliberately, he was really that careless. I don't do this in my own classes, but I've always been in his debt and I am sure to seize opportunities when they arise. – Thierry Zell Nov 6 2011 at 19:38

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@Andrew, continued: I don't tend to make mistakes on the board, but I try to forget as much as possible how to solve specific problems, so that I may very well start out on completely the wrong track. I think this is more valuable than slickly solving every problem being thrown at you, since it resembles the students' own experiences when facing a new problem. – Thierry Zell Nov 6 2011 at 19:42
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Finally, I question in MO I feel qualified to answer!

I am a PhD student in Ireland doing an amount of lecturing. As a first remark, I am lucky in the sense that undergraduate maths was never especially easy for me and therefore I empathise with the average student. My second remark is that I hope for a career lecturing in the Irish Institute of Technology sector where the role in primarily teaching as opposed to the university sector where research is the primary role. Hence I am acutely interested in the skills as a mathematics teacher.

The second half of the answers here are closer to my philosophy than the first. A particular distinction must be put on the classroom environment and facilities. Regardless, my first instinct is that slides alone is sub-optimal.

The alternative to this is to produce everything on the blackboard. I did this last year in a differential calculus module (the students were maths studies --- by and large headed towards a career as "high school" mathematics teachers). The emphasis in this course is to convey to the students that although differential calculus is a relatively intuitive subject with the motivation coming from geometric concerns, as mathematicians we must also be rigorous, logical and precise in our thinking. Hence, we are not merely making a series of calculations and passing exams --- we must understand the content. When I wrote blackboard after blackboard of notes, the students did not have any chance of understanding the material. While I am a fervent believer that exercises and reflection are the best way for a student to achieve this aim, I am reminded of my undergraduate experience where certain obstacles lay in the path of me putting in this work and luckily my presence at lecture-time was sufficient for me to grasp the general theory and progress (eventually with first class grades) despite less than exemplary exam results in previous years. Put simply, ordinary students do not have the faculties to take down written notes and consider the important comments of the lecturer in real time.

However, slides do not work because mathematics is not a spectator sport (not a cliche when the average student is first interested in passing exams --- its is the goal of the educator to transcend this). It takes a superlative lecturer and a cohort of motivated and enthusiastic students to assimilate a lecture purely by ear. At least once I had a lecturer of this standard but I would vouch that were engineering, scientific or humanities students subjected to his fantastic delivery and questioning, they would simply fall asleep. It is a curse but a fact (among my students at least --- none of which are Math majors), that the average student does not have that aptitude to bask in such splendour.

My compromise, therefore is the very similar to what has been suggested above. I produce a set of notes (available soft-bound in a local printing house), with gaps which we fill in during the class (I print the notes onto an acetate sheet which I project onto a screen and can write on with a marker). All the theorems are writ-large, and everything else is teased out per a blackboard with suitable prior fillings in to both give the students a sneak preview and for the practical reasons of properly spacing out my scribblings. Does the need arise, I can put more complicated graphics in this set of notes. Today we introduced implicit differentiation and I projected this Wikipedia page list of curves onto the screen and this was but a two minute interlude.

The issue of students looking ahead was served by a motivation at the start of term (we are studying continuous and differentiable (smooth) functions. We draw a picture. We translate these geometric pictures into an algebebraic ones and never lose sight of this fact).

I have covered more content this year than last using this method, the first continuous assessment results showed a marked improvement and I am ahead of schedule despite being able to allocate a lot more time to comments and explanation of subtleties.

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It also depends on how do you think it is the best for your students to learn : By listening (hopefully carefully) to the course, and then reading notes you'll provide them, OR by letting them write themselves the content.

I don't like to much the first option, certainly because I've not been used too, and I believe it is a huge advantage to write yourself everything at the moment, because of obvious memorization advantages (it was important for me to have my own notations, a kind of taming procedure) and, once you read your notes again, you usually remember where was the parts the teacher got enthusiastic.

Considering then the second option, it is for me an evidence that blackboard win :

• you give the time to the students to write since you do it yourself
• the statements stay longer (at least if you have enough blackboards, or just keep the main Theorem on !)
• there is more interactions content-author-students
• your eyes are not constantly dried by this terrible white light
• it allows improvisation
• it is more classy (personal point of view, I agree)

Against :

• it is suicidal (that is terribly soporific for the students) to NOT prepare a lot your presentation, at least as long as you should spend time one slides
• it requires a good handwriting from the teacher
• its not convenient for drawing complex pictures

My conclusion is then the same than André Henriques !

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Like Terry Tao, I find the transience of slides to be a problem. This is one reason why I stopped using slides as such and began using a single continuous-scroll page for each topic. I lecture from the bottom of the page, so students who are behind can still see the top. (I'm also one of those people who mixes the projector and the board, with bullet points and formulas on the projector and worked-out examples on the board, so I don't scroll down the page very quickly. Fortunately I work in a facility where the lighting allows this.)

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Most of the non-mathematics courses I've taken in college were done with lecture slides, and I have to say that there are a number of advantages and disadvantages to them that actually amount to more disadvantages if you were to do the same in math. The one obvious advantage is that the slides can be posted online, but the problem with this is that it encourages students to skip class. Even those who don't skip class won't take notes (and are sometimes even encouraged to not take notes by the professors), and this would not be good in a math class, because many people feel that copying down proofs from lecture is best way to get a better understanding of them. Also, when you have lecture notes, you can sometimes get nonsense like this. Anyways, back to your point. If your main concern is displaying graphics, you could possibly just use slides for graphics. If you can lecture in a room with a projector screen that doesn't obscure the blackboards, that would be ideal for this. Alternatively you can distribute handouts at the beginning with graphics that you will be referencing.

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Indeed: taken out of context, these slides can be rather puzzling! I don't know if "nonsense" is quite the right term, though... Disjointed, maybe... – Thierry Zell Nov 9 2011 at 17:28

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In lower level math courses at a US university, most students do not want to be in the course to begin with. Providing slides, or even lecture notes, on-line would make it very tempting for them not to come to class. This is a consequence that must be weighed in any decision to use slides. If a student is physically present, you still have an opportunity to reach them. – Chris Leary Nov 9 2011 at 18:05
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