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I teach mathematics at MSc and PhD levels. My preferred method of teaching is old-fashioned: talking and writing on the blackboard at the same time.

Why? Because it has many advantages:

  1. Handwriting: imposes few restrictions on notation and illustration. (Complicated figures I could project from my laptop, but I have no need for this in my courses.)
  2. Flexibility: whenever this is useful, it is easy to 'deviate from the script'.
  3. Natural speed: it imposes a natural speed on the speaker. Preparing slides using LaTeX or PowerPoint and just clicking through them, I find myself proceeding way too fast.
  4. Parallel displays: having several boards available for writing makes it easy to keep some text/examples on display on one board, while writing on another.
  5. Dynamics: referring to information on the different boards allows me to move through the room, adding a more dynamic aspect to the lecture.
  6. Ease: it is a low-tech way of achieving all these things simultaneously with easily available means.

The main disadvantage of this method is that I spend a significant amount of time of each lecture with my back to the audience.

Question: What would you recommend as a means of communication that combines the six features above (most importantly, the handwriting and parallel displays), but facing the audience?

Obviously, a low-budget solution would be appreciated, but my institute is usually pretty generous in investing in technology that improves teaching, so don't let that restrict you!

What I tried: Many things, including writing by hand on tablets (iPads, Digital Paper, reMarkable, etc.) and projecting this in the classroom. Perhaps I haven't found the optimal device for this yet, but it often comes out pixellated, delayed, and less readable than my usual handwriting on paper or the blackboard. Using a document camera to project my handwriting on paper works well, but can project only about half an A4 paper at a time to keep it readable for people in the back of the room and, like other approaches, has the disadvantage of not having parallel displays: it's hugely important to be able to keep definitions, examples, theorems from earlier on for easy reference.

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    Also posted at Mathematics Educators Stack Exchange. – Joel Reyes Noche Mar 20 at 8:32
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    This thread should be thought-provoking, matheducators.stackexchange.com/a/731/689 – Alecos Papadopoulos Mar 20 at 19:15
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    What prevents you from simply doing both? have a set of slides prepared and do them, and if you feel the need, use the blackboard. This allows you to react on questions from the audience quite dynamically, while reaping in all benefits of using slides. And if you notice that what you have on the blackboard is essential, just include it in the slides for the next time you give the same course. This allows iterating on and improving the slides as well. – Polygnome Mar 21 at 0:19
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    @TobiaTesan I think the technology is just now (well, the 3 years ago as I was leaving school) coming into its own. I had a professor who wrote pages under a very sharp document cam, but then pressed a button to save the picture and keep it on one display while moving on to the next page. Multiply by 4 displays and you are on par with the big sliding blackboards of old, with the benefit of saving the notes for later upload (and not having to wipe old chalk off everything). – mbrig Mar 21 at 4:19
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    It seems like most of the answers are simply challenging OP's assumptions and requirements; I am tempted to downvote them all. – Federico Poloni Mar 21 at 7:37

12 Answers 12

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  1. Do not assume that facing the audience is actually better. It depends on the type of content. For example, if you are explaining a diagram (detailed hydraulic system for instance), you want the audience to concentrate on that, not the presenter. The audience is still getting plenty of stimulation by having a live human voice along with visual content. You actually don't want the presenter to distract the eyeballs from the intricate content. (I see this wrong often with recorded talks where the video operator concentrates on the podium versus the charts, or shows both but with inadequate scale to see the slide charts.) If you are doing math (as opposed to history or literature), the content is more intricate and you should have the audience concentrating on the formulas, not the speaker.

  2. Take some occasional time to address the class when that is appropriate. Examples are a "sea story" about test performance or industrial application of the math. But these will be a minor amount of time within a math lecture.

  3. I mean, if you really wanted to, you could write backwards in grease pencil on plexiglass like in an old Navy combat information center. But really what is the point. People concentrate on the status (content), not the writer anyways. The one advantage is that you don't have a body blocking part of the board. I believe there are now electronic versions that would flip this around for you (or you can use an OHP). But still I think you lose a lot from having a lot less screen space than board space (especially with sliding boards). So, the small advantage of facing is at the loss of huge amounts of content physical space.

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    I agree to the extent that I don't want the audience to focus on me instead of what I write. My aim is rather that I want more focus on the audience to check for questions and/or whether too many of them are nodding off or generally going on mental-screen-saver... – Mark Mar 20 at 7:57
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    @Mark Turn around occasionally to check on the audience. I completely agree with the present response. I really dislike writing (or seeing somebody write) on a small pad/paper when it is projected. It is strongly confined and looks like Spock looking into his visor device. When on the blackboard, you make large arm movements, so the whole body movement that produces the writing is obvious to the audience - do not underestimate the effect that it makes, it's far more natural. – Captain Emacs Mar 20 at 8:05
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    w.r.t. point 1: that's fine if you've got a tie-clip mic, but facing the board can make you hard to hear if there's a lectern mic or none at all. Obviously this depends on the size and acoustics of the room as well as your voice, but it doesn't have to be a big room. – Chris H Mar 20 at 9:47
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    @PyRulez My son has toy "spy glasses" with small mirrors at the sides that allow you to look behind you. I guess I'll confiscate them for educational use now. – Mark Mar 20 at 12:20
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    I think your point 1 is simply wrong. Facing the audience is definitely better, for the simple reason that it makes your speech vastly more intelligible. This should be entirely uncontroversial: Everybody who’s ever attended a lecture knows that the way the speaker faces greatly impacts the ease of listening. – Konrad Rudolph Mar 21 at 15:27
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There is an extremely simple solution to this, which is that you stand sideways, with the students on your left and the board on the right. When you turn your head to the left, you're looking at your students. When you write, your body isn't blocking the board, or is at most only temporarily and partially blocking it. When you pause to explain something or ask for questions, you can turn your whole body toward the class and step a little to the side so that everyone has a clear view.

It's not particularly difficult to write in the sideways stance, and it doesn't even really take any practice. When I call students to the board to present their work, I just briefly demonstrate it to them, and they generally can do it immediately.

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    This would be my answer exactly. I am just linking a stock photo in case people are having difficulty visualizing it: 123rf.com/… – Opal E Mar 21 at 18:22
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The retro-projectors where one wrote on the plastic film allowed one to go "back in time" as it were... rewind...

Of course one could lay a diagram on top then remove it etc

So, you can have a single slide with a particular expression or theorem that you need often and drop it on as necessary.

One comment mentions a "camera" to project a document - one that was used to do this is called an epidiascope...

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    Horrible! I remember a prof using such scrolling projectors, and he kept scrolling around, especially back, to remind audience of previous stuff - impossible to follow, makes you lose thread, you force yourself to read/take notes as fast as you can before the next scrolling takes place; after scrolling back to current stage, you have to constantly search again for location you are reading. Do not use! Stick with the good old blackboard where text does not move all the time! Your back to the audience is a very minor nuisance compared to scrollable plastic film. – Captain Emacs Mar 20 at 7:58
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    There is a modern version of this, where you simply write on paper and a camera transfers this to the projector. This way you can show past notes without moving everything else, but the available writing space is still nowhere near a proper set of 6 or 9 boards, even if you have two of them. I know some people who actually prefer them to blackboards, since you can also prepare some sheets beforehand and don't have to budget time for cleaning if the boards are full. – mlk Mar 20 at 8:16
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    Also another note, those retro-projectors are a problem for left-handed people as the students will only see the writing hand instead of the last written line. – mlk Mar 20 at 8:22
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    @CaptainEmacs so the prof used it badly... There are profs who use blackboards badly, or slides badly... – Solar Mike Mar 20 at 9:12
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    These projectors were used at ETH Zürich back in the 90s. It wasn't so bad, but I never managed to not stick on the plastic scroll with my sweaty hands, making horrific messes. I remember Marvin Minsky giving a guest lecture using one, resulting in an incomprehensible scroll of modern art. It got snarfed by some students for publication. They are really best if you have your slides prepared beforehand or if you write really slowly and deliberately. – David Tonhofer Mar 20 at 9:48
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This answer is going to broaden the question asked to instead be, "How can I give the students a better experience, while still delivering content through speaking and hand-writing?"

I would suggest spending dramatically less time lecturing in class. Instead, record the (majority of your) lectures electronically, and share them with your students online. You can use whatever setup you like -- writing on the board(s), writing on paper, writing on a tablet, writing on glass and having the image left-right reversed -- with one or more cameras recording your lectures. You can edit the footage as you like, to allow the parallel displays you enjoy for example.

This has several important benefits:

  1. Students can view your lectures at their own pace, at their own convenience, with the ability to pause, rewind, replay, and fast-forward.
  2. You can use class-time more effectively, by taking advantage of the circumstance of all these people in the same room. Have them talk to each other! They can work on exercises, or go through details of proofs, or work on more ambitious projects. Small groups are most effective, size 3 or 4.

Both of the above benefits are much more important than the ability to see your face during lectures.

  • That sounds great, but what if there are 200+ students, as was the case in my undergraduate lectures. – JeremyC Mar 20 at 22:21
  • 1. OP asks about graduate courses. 2. Even with 200+ students, some of the classtime can be spent in active learning. – vadim123 Mar 20 at 23:24
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    This is not a light change to the course though, because it assumes a lot more at-home study from the part of the students. I imagine that the load at OP's university is balanced on him actually explaining things in class, and not on him asking his students to study videos alone in advance. At the very least, a reduction in the number of total in-front hours would be expected. – Federico Poloni Mar 21 at 17:53
  • @vadim OK I have been to 250+ graduate courses. Yes, you can set a quick task or two, but no real engagement is possible with individual learners. – JeremyC Mar 21 at 22:35
  • +1 for recording lectures: makes life a lot easier if you can't attend classes sometimes. -1 for replacing regular lectures with that: it literally doubles the time students have to spend studying your subject, which is hardly something everyone can afford. Moreover, this means students can't ask questions, correct mistakes, etc. until the next in-person class, which can heavily hinder the understanding of a lecture (what if at t=10min you don't understand a result? so long for the rest of the lecture!) – mattecapu Mar 22 at 10:01
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If the space in which you lecture permits, you could simply put some big mirrors next to or between your blackboards. That would let you continue to write with your back to the audience as you do now while also being able to casually glance at them (or at least some of them) without turning around.

Less unusually, if you're in a small classroom setting you could angle your blackboard a bit away from the class. I've seen plenty of rooms that had a portable blackboard positioned like this before, and it allows the teacher to stand to one side of the blackboard and glance sideways at the class while writing. (I suspect that this is not what usually motivates such setups, and that they're typically done due to a lack of space for the extra blackboard at the front of the room, but there's no reason you can't deliberately use this approach for the visibility benefit.) Of course, in the context of a big lecture hall this would be annoying to the class.

  • Yeah, it really depends on the setup of the classroom. Some have convenient sideboards that are still far enough away from the students that they can pretty much all see all of them; for other classrooms, the students are so close that even the main board is not visible very well for ones on the opposite side at the front. – kcrisman Mar 21 at 14:54
  • @kcrisman It also depends on the size of the class. If you've got a class of 10 in a room made to accommodate 30, you can sit them all on one side to ensure they can see. If you've got a class as big as the room can accommodate, you've got to ensure the board is visible from every seat in the room. – Mark Amery Mar 21 at 14:59
  • Yes, I absolutely agree. – kcrisman Mar 21 at 15:01
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The document camera is, in my experience, the best approach by far.

It has a couple of unique advantages, including producing permanent copies of the lecture notes, both for your benefit when preparing exams and (possibly) for the students' benefit (if you decide to make these notes available, whether to everyone or only in special circumstances). Another advantage is that it's more natural to use than a vertically-mounted chalk- or dry erase-board, so your handwriting comes out more much readable and requires less effort. Finally, it is more accessible to students sitting in the back of the room, where blackboards are generally unreadable due to size, shadows, etc.

The only significant drawback is the need for specialized equipment. But if your institution is able to provide that, you should take advantage of it.

The limited display capability is not a real drawback. It is quite rare that you need to project more than half or two-thirds of a sheet of paper at a time. It simply isn't feasible to talk about that many different things at once. If nothing else, it forces you to keep in mind the audience's human limitations. If you really need to give a "big picture" view, you can always zoom out on the document camera.

While it isn't strictly necessary to make eye contact with your audience, especially since many students will be looking down to take notes anyway, don't underestimate the advantage of being able to actually connect with your students in appropriate contexts. If you are looking out at the class, it's much easier to judge understanding, see those with questions, and just make yourself seem friendly and approachable.

Document cameras allow you to face forward and give a natural lecture, while maintaining all of the advantages of a "chalk talk".

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    Giving an up vote as I think doc cams are the way to go for many things. But in mathematics it is actually quite necessary often-times to allow for people to see as much as three boards' worth of material at a time; I had to today in my linear algebra course. – kcrisman Mar 20 at 17:52
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    Apart from the limited-display concern, which really is a concern for my purposes -- as @kcrisman mentions, it is quite essential to keep a lot of material for reference -- I do agree that document cameras are great. His/her suggestion in a separate answer below to combine blackboard and document camera might be the way forward for my courses. – Mark Mar 21 at 5:23
  • I think OP is quite explicit here on his requirement for a lot of display capability. – Federico Poloni Mar 21 at 17:55
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    Not different enough for a separate answer: I generally use a note-taking app on my iPad, connected to an overhead projector, as a virtual document camera/whiteboard. I get an immediate electronic permanent record, which I can easily distribute to the students. It's easy (with practice) to scroll back to previous "boards" and/or copy-paste-modify previous examples on the fly as needed. I can prepare complicated figures/examples in advance. The screen space isn't as large or as persistent as a blackboard, but on balance it works pretty well. – JeffE Mar 22 at 13:53
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In my opinion complex concepts which involve a lot of mathematical equations are best taught by interacting with the audience and allowing students to write along with the instructor. Many people have suggested here the use of a document camera. How about a transparent-board camera?

I like what Rob Edwards from SDSU uses for his courses on YouTube. For example check out this one. Here is a screenshot:

RobEdwardsSDSU

He uses dark clothing, dark background and a transparent board to write on facing the audience. A camera captures the board content from the front and then inverts the video to project on a display, so that it is in the correct readable orientation.

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    I have tried this for recording segments of my lectures. For classroom use, it doesn't really suit the purpose. Partly due to the "wrong" orientation when writing on the glass in front of a live audience; partly due to the small size of the glass surface relative to most blackboards (of which I fill many during an average lecture). – Mark Mar 21 at 5:14
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I don't have a lot to add to these contributors in terms of nitty-gritty details, but I think that if you can get it having a good setup with both document camera AND chalkboard is best. (Or possibly multiple doc cams.) At least, that is my experience with mathematics in a number of different settings.

When I am in a room that allows for it, I will have certain definitions just continuously on the board, and then dynamic examples (whether computer-generated or paper) on the camera screen. Or you can use the chalkboard for stuff that has to stay up a long time, for which you know people will be copying down anyway - and then you can stand aside as they (hopefully) internalize.

Or, if you have time ahead of time, put needed definitions on some side boards (we have side whiteboards in some classrooms) that you can point to, while keeping attention up front. In any case, you can think dynamic versus static content for the two 'devices' (if a chalkboard is a device).

It does depend upon the type of course. Honestly, for graduate coursework you'll probably be flying so fast it won't matter what technique you use. Given the setup of many rooms in many universities in many countries (but not all of any of the above, of course), I would recommend the doc cam + chalkboard combination as an easily attainable via media that gets at most of what you want in a lecture-based course.

(Whether you want to do a lecture-based course is a different question, but I won't wade into that here.)

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You could write up the main points one butcher paper, then roll them up and put them on the walls. Then you can write out the detailed steps and project them onto the board, then as you finish a section, unroll the summary. This will also provide a stopping point for people to ask questions.

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    This would work very well for certain class types, and I've seen something like it done effectively in education and social science courses, but for a graduate mathematics course it might be tough. – kcrisman Mar 21 at 14:55
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Write with chalk pen/white board marker on plexiglass, video this, then flip the video stream horizontally (in real time) for projection. You'll get 1:1 eye contact, with the students who appreciate that, and whenever they need to clearly see your content they can watch the video. Even better if this is saved for later viewing...

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I had such a math teacher in college. He would walk in, start writing on the board, and not stop until the end of class, then walk out. It was horrible. Everyone in the class hated it. A few complaints later, and he started turning around periodically to see if there were questions, and if the class was keeping up. Spending a significant amount of your time with your back to the audience isn't a problem, as long as it's not all of the time. Just check in with the students, and make sure that they know it's ok to interrupt you when your back is turned (assuming you're ok with that).

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I think one possible solution could be the use of a camera-projector system also often known as a document camera. A common example I've seen in universities around Canada is ELMO your university may consider investing in these and placing them in classrooms around campus.

This system also allows for you to switch to any detailed diagrams or animations you may want to present to your class through your laptop hooked up to the same projector.

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    OP and some of the other answers already discuss document cameras. What does your answer add to the discussion, apart from suggesting a particular model? – Federico Poloni Mar 21 at 17:58
  • Apologies, I didn't notice the other answers at the time of posting – Kunal Mar 23 at 4:55

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