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I have attended several lectures in analysis, linear algebra and "higher mathematics for computer science students" in two universities in Germany. None of the professors giving these lectures provided lecture notes. Therefore the students had to copy what was written on the blackboard while at the same time trying to understand the material.

The sequence of a lecture was usually:

  1. Professor arrives and gives a short salutation and very brief recapitulation of the last lecture
  2. Professor immediately continues where we left off and starts writing proofs, lemmas, etc. on the blackboard while explaining the material in parallel. Rarely he stops to clarify something.
  3. When he has filled all three blackboard, he stops to erase one or two of the blackboards while usually continuing to explain what he just wrote down or introducing the next concept.
  4. He continues with the now empty blackboard.
  5. The whole process repeats from step 2 until 1.5 hours have passed.

I found it impossible to copy the blackboard and at the same time understand the material he was trying to teach. During the few breaks (question or blackboard cleaning) that we had I was frantically either trying to understand the proof or was squeezing my eyes trying to decide whether an index is i or j and was making sure to not copy i instead of i+1 or i+j by accident.

Although he did encourage us to ask questions and did answer them when someone did, the general pace was so high, that it hardly gave me some time to catch up.

When asked why there are no lecture notes, all professors said something along the lines of: "You can learn/memorize better if you hear and write the proofs yourself." While I agree that having multiple input channels helps memorizing, in practice, all that most students could do was to barely copy what was written on the blackboard, there was no time to understand. I felt like the students were "demoted" to human copy machines because this is all what most could manage to do.

Considering all this:

  • Is there really proof that not providing lecture notes and forcing students to frantically copy the blackboard is beneficial for learning math?
  • OR Are these professor just too lazy to create proper lecture notes?
  • OR Is my perception warped and I was in the minority and just too stupid to study math?
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    Is there really proof that forcing students to frantically copy the blackboard is beneficial? No. There might be proof that forcing students to thoughtfully copy the blackboard is beneficial. From my point of view, the pace was the real problem. How did the professors react when asked to slow down? – svavil Jun 5 '17 at 7:01
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    Are you by any chance in the first year of studies? – Walter Jun 5 '17 at 8:20
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    There is a technological fix for this . Bring a digital camera to lectures. Whenever the professor fills a section of board, photograph it. That limits the time when your attention is not on listening and thinking to a few seconds per board section. In the evening, review the photos and make notes if, and only if, you find writing things out helps you learn. – Patricia Shanahan Jun 5 '17 at 10:43
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    Before throwing around offensive words like "lazy", have you ever written lecture notes (for someone else to read)? I have found that creating lecture notes that are suitable for my students to read is many times more work than just writing something I can lecture from, and there are other things I could do with that time that might be more important. – Nate Eldredge Jun 5 '17 at 13:20
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    The "learning styles" thing was debunked almost 10 years ago. Studies have found that while students may have a preferred mode of delivery, that delivery mode does not yield greater actual learning success than other modes of delivery. – shoover Jun 5 '17 at 17:40
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Writing something down absolutely does help you remember it, as more different aspects of your brain are engaged with the material at once. See, for example, this CogSci.SE post on the matter.

The cognitive benefits of engaging multiple channels, however, can certainly be outweighed by the detriments of poor lecturing technique. The problems that you describe having do not actually seem to be about the availability of notes, however, but about the pace of the lecture being too fast and with too little explanation for you to comprehend the material --- writing things down slows you own partly because you have to be thinking about the material and summarizing as you go, rather than just letting it wash over you.

From what you have written, it is not possible for we strangers on the internet to distinguish between two common cases:

  • The professors you've had were going too fast and not explaining enough to teach effectively.
  • The pace felt too fast for you because you had inadequate preparation or some other issue. For example, a dyslexic student would likely need a slower pace.

You may be able to tell which it is by seeing whether most other students are experiencing the same issues as you. If they are, it is likely a problem with the teaching; if they are not, it is more likely the problem is on your side.

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I think your three points for consideration may all contain some truth. Let's see.

  • Is providing lecture notes beneficial? According to my own experiences (as lecturer), there is no clear-cut answer. In an ideal world, lecture notes should be very helpful and most students appreciate access to lecture notes, but providing them does rarely improve the exam results (though I have only anecdotal evidence). Presumably, this reflects the fact that students tend to switch off from attending the lecture with their full concentration, knowing that they could rely on the lecture notes as back-up. But this is a fallacy, because once they are no longer on top of the material, they will be unable to follow subsequent lectures.

  • Are these professors just too lazy to create proper lecture notes? Almost all material taught to undergraduates in mathematics is extremely well covered in many textbooks (often generated from some lecture notes), often much better than lecture notes can do it. So why should the professors take the time consuming burden to write their own notes? What they should do is to recommend a number of textbooks.

  • Is my perception warped and I was in the minority and just to stupid to study math? My first weeks a Uni (also attending a first-year math lecture in Germany) was quite a shock: the speed at which the material was disseminated was completely different from what I was used from school. So, welcome at Uni! The difference is mainly that at school you are given all sorts of explanations and examples, while at Uni you must look after yourself to learn the stuff.

Advice: always keep on top of the material. Nobody expects you to fully understand the content of a lecture then and there, but you must make sure that you have understood the content of the previous lecture before you attend the next one. IMHO, this is the single most important rule for studying math-related subjects at Uni. You should go over your notes, complete them, compare them to textbooks, and do the course work assignments as soon as possible after each lecture. You can do all this on your own, or in a group of peers, but don't let it slip!

----added in edit----

To answer your further question whether writing down the proofs improves your learning: Yes most definitely. And the professors will have had the experience that by providing lecture notes the students actually no longer write out the material (at least not to the same degree).

  • While I appreciate the helpful tips, the core of my question is whether the statement made by the lecturers, that copying the blackboard would help me learn math. I understand and agree that it would be redundant for them to provide lecture notes for undergraduate subjects but then why don't professors simply advise you to come to the lecture and listen and understand. It seems that all three of them were convinced that copying the blackboard was an integral part of learning math. – problemofficer Jun 5 '17 at 8:51
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    Getting back to the core of your question, it’s very difficult to say if that statement is universally true, but it seems to be a widely-held belief, and t’s probably a sound generality. I know more than one math professor who tried using slides but found that students as a whole didn’t seem to learn as well when they weren’t copying from the board. In math in particular, the step of writing things down seems to be helpful to a good number of students. RE: "During the few breaks .. I was frantically trying to understand the proof...” When the prof asks if you have questions, ask something. – J.R. Jun 5 '17 at 9:14
  • @problemofficer - You could add the reference request tag to your question, and edit out some of the context stuff, since it's getting people off on a tangent from what you are asking. At this point it might be easier to start fresh, though. If I were you I would ask on Academia Meta whether it would be better to edit the question or start fresh. If you need help editing out the stuff that's getting people off track, you can ask for that at Meta too. – aparente001 Jun 7 '17 at 20:02
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I'm going to add to the other answers that it is a very German thing for professors to provide lecture notes, and for students to expect them. I have never really understood this practice, because it requires professors to write lecture notes for something for which almost certainly that already are excellent text books.

There may be other shortcomings in the professor's teaching, as the other answers already suggest, but I think not providing lecture notes is not one of those. He/she probably pointed you at one or two textbooks on the subject -- go to the library and check it out, use it in place of lecture notes instead!

  • Many American classes have lecture notes as well. – jakebeal Jun 6 '17 at 11:07
  • No doubt, but many professors teaching undergraduate courses in Germany have essentially a textbook of their own for their students. I've rarely seen this in the US, unless someone is intending to eventually make those into a book proper. – Wolfgang Bangerth Jun 8 '17 at 19:15
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Not really a direct answer to the question, but extended comments on the context:

As other answers and comments have noted, to make publicly-presentable lecture notes requires much, much more effort than the kind of sketchy notes adequate for an experienced person to give coherent lectures. Thus, given the priorities of not only faculty themselves, but of deans and funding agencies (which can affect those of faculty), it is not surprising that faculty would choose not to invest their time in polishing lecture notes.

Also as noted, "frantic" note-taking is not so useful, but with a more positive adjective it does (in my own experience, and in my observation of others, over several decades, in mathematics in the U.S.) create a more intense engagement with "the moment".

As recently (!) as 35 years ago, it was not so easy to create typeset lecture notes, for technical reasons, apart from the possibility of photo-copying hand-written notes. In particular, especially for more advanced courses, the tradition of skimpy notes-to-self amplified by the lecturer during the lecture, written down by students as their main source, is still vivid in senior faculty's minds. And, as above, endorsing this does rationalize avoiding the effort that writing-up serious notes would entail...

I do claim that thinking in these either-or terms is misleading, though. I started thinking about this in 1974, while attending lectures of a very eminent mathematician: he obviously had very carefully prepared his lectures, writing everything out in long-hand (=cursive) on lined paper, and very carefully copying it onto the blackboard, without saying a word. :) Eventually, when typesetting mathematics became much easier, I realized that written material and "live" material (with blackboards or whiteboards) are very different mediums, and do not successfully/usefully imitate each other.

That is, the typeset notes or text should be one thing, and the lectures another. Lectures as a vehicle to impart written material fails... Written material to convey personal excitement, or to interact with audience questions... possibly steering the whole course of the discussion... fails.

So I do attempt to provide respectable notes for my courses, though they do not pretend to echo my lecture/discussions, and in lecture I do not necessarily pretend to "follow" the notes, since I pointedly attempt to react to questions and general audience reaction (which, obviously, notes cannot do).

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