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I am writing this from the perspective of a mathematician. I am wondering whether there is any research on the standard mode of instruction.

Mathematics lectures, at least in my home country, typically work as follows: the prof starts writing on the blackboard (or whiteboard) while telling about the notes, and students copy down the notes from the board.

90% of the lecturer's speech is reading out the notes aloud while facing the blackboard. Depending on the level interaction, there may of course be questions to the audience, but mainly it is reading out the lecture notes.

Most students aren't really listening because they are busy copying down the notes from the board. You typically see them turning their back and forth between the board and their own notes. Students may find themselves in "unintelligent copy" mode, just copying symbols from the blackboard without any comprehension of meaning.

Irrespective of your style of instruction, I think this is a terribly ineffective concept of teaching right from the start. Individuals have told me that they believe in some sort of "subconscious diffusion" of the lecture material into their students' minds, even if their students are just in "unintelligent copy" mode. I don't believe in this and find this rather counter-intuitive.

All in all, much of the typical way of lecturing looks like a giant waste of time to me. I can only imagine that the most gifted students may benefit from this, and perhaps that is the reason such a mode of instruction is pursued.

Has there been any research on this mode of instruction and its educational benefits? Has there been any research on that "diffusion" hypothesis?

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    It should be emphasized that the question here is Has there been any research on.... Answers that contain merely someone's opinion are insufficient, in my view, and should be downvoted. Please stop giving this kind of answers. – Federico Poloni Feb 28 at 17:02
  • Comments are not for extended discussion; this conversation has been moved to chat. Please see this FAQ before posting another comment. – cag51 Mar 3 at 1:21

10 Answers 10

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Prior to electronic media generally and after the start of mass, rather than individual instruction, this was the primary way of teaching a lot of things. It can be a good start to learning, actually. However, it isn't enough, and never was. Learning is an active sport, not a spectator sport.

We learn by actively engaging the brain. Copying down notes is somewhat active. Certainly more active than taking a phone picture of the board once it is full, or downloading a lecture. It is even more engaging than (just) reading a lecture.

However, if you really want to learn, or teach someone how to learn, I suggest a few additional steps.

One is to review the notes after a day or so.

A more extreme version is to take the notes by hand (pencil and paper) and then transcribe them and add annotations that are needed, perhaps preparing questions for the instructor or yourself to answer.

One is to extract and write down the most important message in a lecture. Or maybe a couple of them. This can be done at the immediate end of the lecture.

One is to ask questions during the lecture if a step in a proof isn't clear to you. (Yes, instructors make mistakes).

One is to review the most important points of the previous lecture just before the next one.

But the most important aspect is to do some kind of exercises based on the lecture. Perhaps these will be assigned. Perhaps they are available in a text book. Try to get some sort of feedback on your solutions.

Another is to start a discussion group with a few students to review the lecture material and update the notes.

But the real key to this is that you learn by actually rewiring the synapses in your brain and that takes repetition and reinforcement. To assure that you are reinforcing the correct ideas, you need feedback.

Note that watching lectures (say on the internet) without active participation is almost completely ineffective for most people. Especially for long term retention and the ability to put the knowledge to use.


For some of the biology behind learning that suggests reinforcement and feedback see: The Art of Changing the Brain by James E Zull


Note also that, as a professor, I've had to explicitly teach this to students and force some of the activities. For those that learn easily they may not seem to be needed, but eventually things will get hard and having good practice can save you.

For example, I'd end a lecture a couple of minutes early and ask for "the most important lesson of the day", expecting that the students would have note cards with a few ideas.

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  • I think this is a good take for most lectures. In particular, I think trying to work out what the key points or, or important parts is at least partially an active activity. Things are turned from words to meaning, and then back into words on the way back to the page. However, does this transfer to math? Presumably it is important to get down all details of all parts of a proof, and in the processes, little processing goes on - I know that when I do this it goes straight from symbols on the board to symbols on the page without passing through meaning. – Ian Sudbery Feb 27 at 16:51
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    We learn by actively engaging the brain. Great that somebody is saying that. In my student days, we all learned the old-fashioned way and the lectures were slow enough for me to simultaneously a) listen, b) write down and c) comprehend and reprocess the material (so I was anticipating what's coming next except for a few really surprising twists in the proofs). That mode required some effort on both my and professor's side and I asked questions immediately when I felt somewhat lost, but otherwise I didn't need either an eye contact, or any of modern flashy props; the flow was perfect. – fedja Feb 27 at 17:23
  • Note-taking engages three sight, sound and feeling which is surely no less useful than the more modern method of teachers using hand-outs. What's in between? – Robbie Goodwin Feb 28 at 22:34
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    For me, the act of physically writing the notes was key when I was younger. That was what it took to get the content into my brain. I then could remember it, visualizing the page it was written on. I then never actually went back over the notes to review - my time was better spent (to me) looking at other problems, particularly how other authors than our textbook approached things, to compare and contrast. There is research showing that actively writing notes results in better learning that having handouts given to you. – Jon Custer Mar 1 at 3:56
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    Does not work for some of us. I could either listen to and reflect on the lecture or write it down: getting both was not happening. In math it had to be thorough since a missing step could later leave me lost. I had poor handwriting so it was extra effort just to make it intelligible even to myself. Nothing left over for learning. I would instead listen/reflect/ (sometimes ask questions) and then as far as the notes .. hope for the best .. Translation? Read the text - which rarely covered all of the lecture material - and/or hope I could cheat in a sense by copying a friend's notes. – WestCoastProjects Mar 1 at 4:26
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I find this lecture style to be basically useless. I hate giving lectures in this way and students hate taking them. With 500 people in a class there is no real alternative (I'm including zoom versions of this same thing). I also find purely flipped classrooms to be basically useless. Maybe if I ever taught a class with 5 people it would consistently work. I don't know. My experience has been hybridizing lectures, flipped classrooms, and student-driven presentations is a good way of dealing with classes that have 10-40 people.

At the end of the day the lecture style that the professor is comfortable with is always best. Nothing's shittier than a poorly run classroom - you're better off with just lectures than some flipped classroom where little or no thought has gone into the course design or execution. Some students will do better in each type of classroom and some will do worse. Somewhere there's a student with no imagination but a good ability to memorize things that they're told who loves lectures.

There has been a ton of research on this topic and I find very little of it valuable.

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    A flipped classroom with students working in pairs can work, and actually be quite effective as well as efficient. Students don't get stuck as often and require the instructor to get them unstuck when working in pairs. – Buffy Feb 27 at 16:09
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    "At the end of the day the lecture style that the professor is comfortable with is always best" No, that's always the laziest. The best lecture style is that which best allows the students to learn. That may require the lecturer to put in some effort, but that's called "doing your job". For myself, my time at uni was 100% "sit in a hall while a lecturer drones at you from 20 yards away", and we had two lecturers who were literally incomprehensible due to their accent. All the lecturers were entirely comfortable with this - and all the students were entirely let down. – Graham Mar 2 at 11:32
  • @Graham I don't disagree with you, but that process of changing classroom style has its own costs. There's gonna be a couple years where students are guinea pigs, and it's gonna suck for them. – user133933 Mar 2 at 17:47
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    @Graham The key (to any style) is sufficiently slow pace and constant feedback. Accents are not that bad. We had a couple of professors born with speech defects and, if one put some honest effort into listening to their pronunciation, it stopped being noticeable by the end of the second week. 20 yards away is a problem, of course, but who prevented you from sitting closer? My experience is that the front rows are always half-empty and students have some herd instinct to gravitate toward the middle rows. Learning requires some effort on both sides. – fedja Mar 3 at 3:10
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For what it's worth, as an undergrad I made a point of copying down the notes from the board along with margin notes whenever the professor said something that sounded wise. These margin notes often turned out to be extremely helpful. For example, scanning through the notes I made from Probabilistic Combinatorics:

  • Important ideas which for the sake of speed weren't written down. "Idea: induct on the number of blocks", "this bit is by Lipschitz continuity"
  • Structural facts about the material. "This is the same as Theorem 4 but with a different proof"
  • Relative importance of the material. "This lemma is silly", "The Hoeffding-Azuma inequality is the bread and butter of every combinatorialist", "we could have 48 or 72 lectures on this"
  • Asides to help us get more intuition. "very strong concentration of measure", "This is the only possible definition"
  • Notes about my own confidence. "should we interchange $p$ and $q$ here?"
  • Notes about the course. "Examiner's Favourite"

The lecturer tells you a lot more than they write down, and a lecture is an extremely useful and highly scalable opportunity to obtain not just knowledge about what is in the course, but a bit more of the culture of what you're learning. (I do strongly believe that students should be taught how to take lecture notes, and that if they do not know how, the experience can be much as you describe.)

See Prof Körner for an entertaining paper on the subject, which I think should be required reading: https://www.dpmms.cam.ac.uk/~twk/Lecture.pdf

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  • It is reassuring that Körner addresses exactly the same concerns about lecturing. – Ambicion Feb 28 at 14:30
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    (+1) Although this doesn't answer the question asked, I think it gives very good advice for what should have been asked. I often had the textbook in front of me during lectures, and if there was a useful observation made (e.g. the theorem continues to hold if the T_1 topology hypothesis is weakened to T_0, this proof that open connected subsets of of the complex plane are pathwise connected actually shows they are polygonally connected, etc.), then I often wrote it in the margins of my textbook. I learned the phrase "an epsilon/3 proof" in lectures, which I don't think I've seen in a book. – Dave L Renfro Feb 28 at 17:34
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Has there been any research on this mode of instruction and its educational benefits? Has there been any research on that "diffusion" hypothesis?

As you start with the request for academic research in your text and finish on that same note (not in your question title, BTW, hence plenty opportunity for opinionated empirical personal evidence reports :-)) ) here's what a run around the research net delivered:

Math specific stuff? ... Not so much

There is some published material around college level math ed, but that nicely clusters with the 'effective teaching & effective learning scholar search query' mentioned further below.

And then... on to "note taking" and the "subconscious diffusion" hypothesis

There's plenty on note taking in various general or non-math scenarios, but most is about doing it most effectively, e.g.

Then on to the "subconscious diffusion" hypothesis: despite my personal objection to note taking effectiveness per se (I personally found it had negative impact, so I came up with a system where I didn't take notes at all when I was moving around in higher ed -- this was before the advent of mobile phones with great cameras, BTW, so no other recording mechanism was used) there is apparent evidence of its relative effectiveness in learning, based on EEG scans, etc.:

but then I could NOT find papers on the comparison of note taking vs. no note taking at all. (Only some peeps mentioning note taking is considered more "active" than listening attentively -- I disagree, but this is very different for different people, so my position is not applicable to the general student public.)

A counter-search for research results which might conclude that listening is more effective than note taking, or rather that note taking is detrimental to listening, has not been found yet.

I say yet because the plethora of publications focus on improving your note taking skills -- some universities even publish handbooks about effective note taking! -- and I do not expect much research into that specific subject as "note taking" seems to be the agreed upon (undisputable?) common denominator in higher ed.

An evening of meta-research like this might lead one to the cynical conclusion that the world has already lost its youthful "ungrounded idealism" and collectively gone through the 5 levels of trauma processing, arriving at the very mindful accepting stage where mediocre and inferior teachers are to be expected to be present in bulk, this majority due to the huge demand for and meager supply of teachers everywhere, hence obviating the strategy where the resultant problems are better carted off onto the backs of the students ("here's some life's experience for ya, can't receive it too early") and consequently research tracks the hurd, focusing on how to get that bunch of homo sapiens through the ordeal as optimally as possible. Which leads us to...

[Edit:] ha! found some mention at least of research including not taking notes:

  • How Much Mightier Is the Pen than the Keyboard for Note-Taking? A Replication and Extension of Mueller and Oppenheimer (2014) Kayla Morehead & John Dunlosky & Katherine A. Rawson (2019) https://link.springer.com/article/10.1007/s10648-019-09468-2
    (and no need to pay Springer 38 European Pesos if you use your brain ;-) )

There's also this (and more, once you've hit these in Scholar):

Another bit of research that may be of interest here and could possibly assist in adapting your studying environment/strategy (I assume the teacher is not malleable in his behaviour):

because let's get real: ultimate this is not about note taking good or bad or whatever, but achievement in education, i.e. making it out of there, with a degree if possible, and then on to the real stuff. (and more interesting people -- for you, at least)

Teaching effectiveness research

Ho boy! Drop those first two words in Google Scholar or Bing Academic and you'll be treated to a lot of research being done on improving the teaching effectiveness. (And there's also "Learning Effectiveness" results, though you might want to filter out the AI/Deep Learning noise there)

Little attention to math specifically, but the research spans the entire gamut from preschool to higher ed and adult ed, generally ending up with the conclusion that "classical education" isn't exactly bad but can be improved in various ways, improving: Attention, Variation, Interaction.

Regrettably nobody I checked made the effort to get really nasty and select for horrible teachers as a baseline, though there are some blogs about "inefficient teaching methods". I would like to mention this paper:

and I'm withholding the rest of the title ;-) , because from your description of the class experience there's plenty overlap, despite this being a report about a zero(0!) percent pass rate in schools in Zimbabwe and what possibly went wrong over there. Perceived low morale from the teacher's side being a major common factor with Zimbabwe here, for his teaching methods as you describe them seem not so much "classic" as more "curmudgeon" (facing blackboard, questions to the audience instead of from the audience, etc.)
if I was feeling more compassionate tonight, I'd say the man is tired and probably does not like being in front of a class either. It happens. A lot. Who knows...


Enough fun now, it's a large field of research and a lot is happening. However, as with all endeavours that are very personal, the overall take-away of a lot of the results is "it depends" (on the student, the teacher, the environment, and so on).

Oh-kay, but did anyone compare this prof's modus operandi with anything else in edu land?

You wrote:

All in all, much of the typical way of lecturing looks like a giant waste of time to me. I can only imagine that the most gifted students may benefit from this, and perhaps that is the reason such a mode of instruction is pursued.

Before I attempt an answer:

A note about the "most gifted students may benefit": wrong.

When you are gifted, even a little, you do not benefit from regurgitation like that, because you already picked up where this was going from the books and syllabus that go with the course. All you need is a couple of hints along the way when you get stuck somewhere, the rest is just pedal to the metal at your own pace.

Fundamentally, "gifted" you has three options:

  • Either you make sure you get some extra intel during class that's useful and interesting to you (extroversion helps here ;-) ),
  • or you do as I did and skip class entirely, going elsewhere for your education: meanwhile, read (nay, grok!) the books, grab/trade the notes off the others where you deem this necessary afterwards (to make sure you don't miss any additional info that inadvertently happens to be disseminated during class by the sub-top teacher and might otherwise hit you in the neck at the exams),
  • or you become severely depressed, because you don't receive the necessary amount of input to keep your brain engaged. (I've seen those and generally this tends to spiral towards heavy alcohol abuse. Total drop-out, ending up doing something completely different, is your option when you're lucky.)

(yes, those three options are logical-OR-ed together. There's no exclusivity requirement there. ;-) )

So, please, rethink what it might mean to be gifted in a setting like you describe. If it's already "wasted time" to you, how can it be fruitful to you at all when you are gifted?
Unless your prof took off like a moon rocket at the start of the semester and you are still sitting there flabbergasted only able to hurry your notes, then, okay, maybe he's running at a speed that a gifted person might appreciate. Still, the way you're telling it his interaction with the class leaves things left to be desired and you didn't sound like you're out of your depth yet re the material discussed in class either, so...
nah, doesn't sound like he's in it for the smarties only. Sounds more like he's counting down to retirement or meeting with his buddies instead of having to gab with all the collected bloody youth in there to claim the paycheck. That's being depressed at an older age.

Now about an answer to the "waste of time" from research:

If we can agree that the basic strategy of your prof is using the transmission instructional model of teaching (he disseminates his knowledge to you folks, instead of leading you on a path of discovery), then there's plenty researchers have to say, and, again, ignoring his specific qualities as an educator here, but merely focusing on the various strategies, then I 'read' the research as, once more, "it depends". But that's me and an evening of very interesting publications. You can research this and come to your own conclusions.

Let me approach this one from a different angle, where I come back to your own stated desire to have a mathematics focused answer: I serve and aim to please. :-) This one is about your future competition in the world and very interesting on various levels, with lots of references to material that's much closer to home geographically:


Aftermath

I wrote this while doing the research and it was fun. While I did not really address this from the perspective of a mathematician (after all, where's the hard logic and the math in here?), I wrote this more from the perspective of an educator interesting in education research, as at least there we might have some rapport as you wanted to hear about scientific research regarding your issues. (The professionals in the edu field have a yearly conference (ICED); this is serious business. If you are interested in this area of research, you might want to check them out.)

While you (OP) may be bothered about the note taking, etc., in my opinion, you (OP) have an over-arching problem causing all this, which might be stated as this question:

How do I spend my student days more optimally? How do I get more out of this endeavour?
[Because right now, I feel like I'm wasting my time and having to adhere to bosses (professors) that do not enthuse me at all.]

Asking that question (and trying to answer it) might lead you to more useful conclusions and strategic and tactical choices.

Without going into it further, here's a few questions for you:

  • if note taking is such a stressful operation, why not distribute the task?

    Have a few people take "official" notes, while others can copy them. Thus they are free to pay more attention to the verbal/interactive part of the lecture, entice the professor into interaction instead of droning on, etc. and make only notes about those few bits they feel very strongly about. Then get together and merge notes or study together. (What researchers mention as more effective note taking by not acting like a photocopier all the time, but digesting the material and only noting those bits that were surprising or otherwise noticeable hints to drive your thought processes. Haven't seen mention of any "team effort" in those papers to game the system like that though. A pity.)

Of course, this requires teamwork and a working team is a long term mutual contract, for mutual benefits: the "note takers" are not to be "used up"! -- personal experience is that some people are good at note taking (seek out the ones who score B/C and are happy with an A once in while; the rare summa cum laude folks write notes only Bletchley Park could decode in their heyday so they're effin' useless for a scheme like this) while others are not good at it, so don't step into the simplistic "equality" trap by rotating the note taking. You'll fail horribly, all around.
I got things done by teaching other students stuff they had a hard time with and they provided me with their notes in exchange. Worked like a charm for years. The "note takers" in the group can gain from the discussion afterwards, when you can explain stuff they might have missed or glossed over while making the notes. Leverage your individual skillsets. Experiment. You'll be surprised. (I discovered that teaching was bloody hard (and cool!), as it goes way beyond verbalizing your own comprehension. Taught me a lot extra about the material, people, communicating information and myself. Never would have discovered that if I'd stayed in my box.)

  • have you asked the prof for lecture notes? If those exist, that can save a bundle on note taking effort too.

  • why not be better than my old self and take the prof aside and ask him how he feels? No critique, but are you interested in maybe trying something different next time, sir? Like maybe try a Q&A? Let's try something different, shall we? Not permanently, but once in a while, you know, to keep things interesting? Seek the joy and wonder, try to find it. (That'll be a tough job and needs someone emphatic with the prof to help kick it off, but things can happen. Positively.)

    Of course, I can remain cynical and pop in with

    but may I be idealistic for a mo' and suggest you try to counter that one by showing some surprising collective initiative? I hope you do!

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    Another option is to actually do your homework/self-study while sitting in class and keeping an ear open for any actually interesting parts of the lecture. If you can get one lecture ahead then it's not too hard and as a nice bonus you look very diligent, sitting there writing away. – user3067860 Mar 2 at 17:51
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I think this learning style is a fundamental misunderstanding of what "taking notes" mean.

I am one of those people that has a lot of problems paying attention without taking copious notes. This was true when I was a student, and it is even more true now that I am a researcher attending seminars and work meetings. The trick is that I never (or almost never) just copy down what's on the blackboard or what it is said out loud. I summarize and reinterpret, often trying to anticipate what the speaker is saying. What I write is not what the speaker says, but the understanding I get. Essentially I am retelling in real time the presentation to a third party (the piece of paper). I am so used to do it that I find very hard to concentrate on something without pen and paper to do this process.

Now this is very tiring and even now, with many years of practice, I cannot do it for more than 1h30 without a pause. Moreover it is a skill, and one that's non trivial to acquire (it took me many years in middle and high school to become proficient in it).

The problem, the way I see it, is that many students that could benefit from this learning technique simply never learned it, and use as a fallback mode being the "dumb transcriber" of what the professor says (which is frustrating and pretty much useless). I am not sure what could be done to fix this or what the root cause of the problem is.

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    As a student, I'd say the root cause of the problem is when I can't keep up with the speed of the lecture while thinking and taking notes. And a simple way to counteract that is to provide the slides/notes before the lectures, so I can simply screenshot and paste the relevant parts to my notes and add my own notes. Typing formulas takes way longer than that, for example, leaving less time for thinking along. – lucidbrot Feb 28 at 22:00
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    @lucidbrot That's also why I strongly dislike slide lectures -- blackboard forever. Writing stuff on a blackboard forces the lecturer to move at a reasonable speed. Slides pushes you to go through them super fast. – Denis Nardin Feb 28 at 22:03
  • I guess that depends on the lecturer. My analysis prof usually filled at least four full-room-width blackboards per two-hour lecture. But you make me realize that it's not so much about the medium than about the time the students have to think (and maybe discuss with the neighbour. I always found that very helpful, at least if my friend understood it) – lucidbrot Feb 28 at 22:13
  • How does ones "reinterpret" a mathematical proof that is just a string of algebra? – Ian Sudbery Mar 1 at 20:43
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    @IanSudbery I've never met such a proof in my life as a mathematician. Usually proofs have ideas behind it (even those that can be summarized as a string of computations), and writing the ideas is a lot more important than writing the computations. If the proof is trivial I just write "trivial". – Denis Nardin Mar 1 at 21:14
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Most of the deeper learning happens after class when working on homework, rereading the notes, asking yourself questions, etc. But that doesn't mean that lectures are useless.

In a setting where you have 50 minute classes and a lot of content to cover, what other options are available? I think the best scenario is if students come into class already acquainted with the new content.

If lecture notes are made available before class and students read them, a lecture could be oriented to helping the students understand the notes more thoroughly, and that would be a better use of time. This way students are actually paying attention to the content, not paying attention to "getting everything down." But it requires that the students actually prepare for class, which they might not do, and that lecture notes be prepared well before class, which the professor might not do.

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This is just my take on this as a student. For me copying what the teacher writes is a way for me to force myself to think about the material. When I'm not actively writing my mind goes like 'yeah I understand this' but when I write along I have to actually understand what I'm writing down. I don't even use my notes for studying afterwards it's just one way to process the information.

This method that the teacher uses can be a good method but only under certain circumstances. If you have a bad teacher at some point you will lose track of what the teacher is saying and the remaining time of the lecture will be a torturous stream of seemingly unrelated facts.

If you prepare for your lecture you have seen most of the material already and since you're on your own you can spend more time on the things that are harder for you. The lecture becomes a repetition and since you have seen the material at least once this gives you a chance to process it a second time. You need less effort to understand everything so you have more mental capacity to really think about what you're doing. To be honest I haven't been doing this too much myself but I hope to do more of this in the future.

So my conclusion is that if you pair a good teacher with preparation before the lecture this method can be really effective. It might not be optimal but it's more than good enough. If you have a bad teacher and you're not prepared these kinds of lectures are almost a waste of time.

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It's an oft-recounted fact that nearly all pedagogical studies done on this matter show that students who take notes at lectures perform better than those who do not. Refer to 13:20 in this intro pres. In that video the professor refers to the combined effect of using both sides of the brain when listening and writing. Whether or not a lecturer benefits from writing the notes is not something that I can be definitive on. Obviously, it will depend on lecturer fluency: less fluent lecturers will want more time to explain things and hence the increasing use of pre-prepared overhead projection notes since the 70s. But this benefit to the student presupposes that the pace of note-taking enables concurrent absorption of the more elaborate explanation being provided by the lecturer. In practice, with certain subjects (math in particular), with large classes and interrupted views at the far end of the theatre, a display board behind the lecturer, bad timing between the explanation and the notes, etc, this may not always be the case.

In the latter cases, given that the lecturer is working from (i.e. essentially copying) an existing set of notes, it might be better for all concerned that the notes be printed with generous margin space and distributed among students. The student can then focus on:

  • Understanding the matter being presented
  • Composing his/her own syntheses of point on the printed notes' margins

However, just because everything seems clear enough in class, this doesn't remove the need to re-read and regenerate more fluently these classes in home study - as well as the obvious task of doing exercises based on them.

Overall, I think it's hard to offer a comprehensive rule on this question of the "best way" to lecture and to learn from lectures. If classes were handy (< 30?) and the lecturer a fluent speaker and fast writer and conscious of his/her position w.r.t. the board, then maybe the traditional way would be best. But then students have widely varying ways of looking at things and a range of preliminary knowledge. I feel that this is something that each class and their lecturer should discuss beforehand as soon as possible after a course starts. And if most of the class find the initial method awkward then changes should be suggested and taken on board.

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Is hastily writing down the professor's lecture a good way of learning

Ans: No,your life and time will be waste you will not gain deep knowlegde and understanding ,

From Prof Terence Tao statement regarding learning mathematics

It’s also good to remember that professional mathematics is not a sport (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest “score”, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications. For these tasks, mathematics needs all the good people it can get.-

We learn 10 percent of what we read, 20 percent of what we hear, 30 percent of what we see, 50 percent of what we see and hear, 70 percent of what we discuss, 80 percent of what we experience, and 95 percent of what we teach others

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  • Hi Jasmine- I read your answer and did not really see how it tied back to the question. – WestCoastProjects Mar 1 at 4:32
  • @StephenBoesch im saying that one of the best ways to know whether you really understand something is to try to explain it to someone else... – jasmine Mar 1 at 4:36
  • OK - not disagreeing on that but it's still not tied back to the question. – WestCoastProjects Mar 1 at 5:31
  • Reasons for downvote - The answer does not mention the actual question of OP ("has there been research"), nor does it offer sources for its own relatively crass statement, nor does it match my personal experience (maybe irrelevant, but there's that)... – AnoE Mar 1 at 13:01
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Yes, of course there is research on taking notes when the lecture method is used, etc. See, for example (found with Google):

https://www.cultofpedagogy.com/note-taking/

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    That's a very nice site. May I suggest you enlarge your answer by elaborating a bit on what the page says - you don't need (shouldn't) just copy it verbatim, but maybe you can give a short summary. Else, that site might go away in the future and thus the answer may become useless. – AnoE Mar 1 at 12:59

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