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I'm a first year math grad student. As an undergrad, all my professors followed the given text book closely, making lecture easy to prepare for and very predictable. As a grad student, my professors seem to cover material mostly in the book, but in a different order, with different notation, or different content entirely.

The result is that I have essentially no way to prepare for classes. I can only show up to lecture and hope I can keep up. If I want to review, I can only read their handwritten notes. My understanding is really suffering because I can't reinforce the content by seeing it repeatedly and clearly. I know it's unreasonable to expect professors to merely regurgitate a textbook, but this is how I'm accustomed to learning. What can I do to learn material better in these situations?

Edit: Thank you for the replies. The consensus seems to be that lectures shouldn't reinforce the text. They should provide a bigger picture, a linking of ideas from the expert's point of view, and understanding details is the job of the student on their own time. This makes total sense.

I think my issue comes from spending valuable time studying the book, then having the professor totally skip that topic and go to another, never to mention the thing I spent hours studying. My effort is wasted, and I have to expend double the effort to understand what is covered. It seems like the answer is to just learn what I can and not see that effort as wasted.

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    Say something about scale. How many students in a lecture? Say something about how the course is presented - zoom, slides, whiteboard, ... – Buffy Apr 21 at 11:22
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    This might be a dumb question but, why are you reading their handwritten notes and not yours? – Denis Nardin Apr 21 at 15:13
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    This needs a country tag. "Follow the book" is (I believe) an US-American thing. – Marianne013 Apr 21 at 19:50
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    Please don’t write answers in comments. It bypasses our quality measures by not having voting (both up and down) available on comments, as well as having other problems detailed on meta. Comments are for clarifying and improving the question; please don’t use them for other purposes. Existing answers in comments and other extended discussion has been moved to chat. – cag51 Apr 22 at 3:24

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If your undergraduate lectures were perfectly predictable, following the book exactly, and you showed up perfectly prepared to each one, then they were actually completely useless. You could have just stayed home and read the book yourself!

At the graduate level, the best practice is to ask the professor for good references, both at the start of the course, and whenever unfamiliar topics come up. You shouldn't expect immediate full understanding from going to lecture; graduate subjects are too deep for that. Treat lecture as an informal invitation to the subject. It's your job to turn that into understanding, by reading from references, doing problems, and asking questions.

Of course, this isn't an excuse for lazy or disorganized teaching. When a professor doesn't follow a book, they could be doing you a great service, by presenting a carefully thought out, alternative view of the subject which will complement existing books. (That's where many textbooks come from in the first place!) But in other cases, they could simply not care much about teaching, and not have any direction in mind.

I've been in classes of both types, and if you're stuck in the latter, it's absolutely essential to get good references to solidify your understanding. In these cases, don't count on the professor to supply them -- if they aren't paying much attention, they're likely to recommend books they used decades ago which are well out of date, or subpar books that they've only heard of, not read. Ask other students, and try resources like StackExchange. You can find good resources for any graduate course in any subject in this way.

I think my issue comes from spending valuable time studying the book, then having the professor totally skip that topic and go to another, never to mention the thing I spent hours studying. My effort is wasted, and I have to expend double the effort to understand what is covered.

If you're planning on doing research later, then effort spent learning core material in standard graduate-level books is never wasted. You can't know in advance what's going to be useful for your research, and often it will turn out to be the core material -- which is why it's the core in the first place. Many times, I've been stuck on a research problem only to find that the right way of thinking about it was in a book chapter I skipped long ago.

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    If your undergraduate lectures were perfectly predictable... then they were actually completely useless. - This is an exaggeration at best. Things like interaction (Q&A) and presentation matter. See Rota on Church for an extreme example: mathshistory.st-andrews.ac.uk/Extras/Rota_Church – Kimball Apr 21 at 15:49
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    @Kimball that’s a fun (and funny) story about Church, but the sentence you say is an exaggeration is “morally” true. A lecture whose contents can be predicted in advance to a high degree of accuracy is certainly a lot less useful than a lecture that doesn’t have that property. (And knzhou’s answer is a lot more useful than other answers even though — or perhaps because — it contains such an “exaggerated” but morally true sentence.) – Dan Romik Apr 21 at 16:32
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    @DanRomik The utility of a lecture is determined by how well it serves student learning, not by how novel it is. – Elizabeth Henning Apr 21 at 16:41
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    @ElizabethHenning I wasn't claiming lectures had to be novel, just that they shouldn't be perfectly predictable. For example, active learning works because it asks students to solve questions they have a roughly even chance of getting right. That uncertainty in outcome allows the class to both learn (when wrong) and teach each other (when right). But it's not novel at all -- you can buy books with lists of such questions. – knzhou Apr 21 at 16:46
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    @DanRomik I'm not critiquing the spirit of the answer, but the wording of the first paragraph rubs me the wrong way. E.g., in my mind, I follow the text pretty closely for a course like calculus, but I don't like to think my lectures are mostly useless. Perhaps the difference lies in knzhou and my interpretations of the phrase you showed up perfectly prepared: to me this means you showed up knowing what you need to understand the lecture in real time, and not that you understand the entire content of the lecture beforehand. – Kimball Apr 21 at 20:23
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I am a retired lecturer, so here is my perspective.

As an undergrad, you were part of a sausage machine, taking kids from school and running them through a training process where the competent ones could earn a Degree that said they were competent to ply a given "trade".

Grad School is different. In many Unis it is a lead in to research, or if it's a coursework Postgrad, it's studying advanced materials where there may not even be a published textbook. Either way, a student has to show resourcefulness, independent thinking, and determination.

Often the lectures are led by people talking about their own research area. Sometimes the lectures are actually a selection tool. Students who do well may become preferred candidates for research places, assistant roles, and the like. If there is a tutor or assistant then they may be a student who is supervised by that member of faculty.

I think it's reasonable to request lecture notes, but don't be surprised if they consist of "Discuss Chebyshev Polynomials of the first kind, second kind, multipole expansion, and applications". It's also reasonable to request references, but again, don't be surprised if you're told to find them yourself.

After all, after this you should be able to do independent research. And publish it.

To return to the question, my advice is basic student stuff. Sit at the front. Take notes by hand in the lectures. Use a recording device to capture the words. Ask if video is available or permitted. Followup with those handwritten notes. Connect with the other students. Do any problems that are set. Attend every class.

And speak to the lecturer, asking questions about the material, not about learning issues. You may get the brush-off, but maybe you'll be rewarded by the academic seeing you as a serious and interested student. Everybody likes it when somebody is interested in their work.

[ Edit ]

Thanks to @BillNace and @ElizabethHenning for pointing out improvements to this answer.

In my experience there are two kinds of academic: those who care about teaching and those who only care about their research. I suggest the OP broach only questions on the material, so that they can get a sense of the academic's approach to students. If it seems positive then that is the time to broach these concerns.

If not, then the Faculty Office may provide a service, after that the University student support service or student union are the next ports of call. Every University that I have had anything to do with, even just to visit, has had a student support service. Student unions or associations are ubiquitous, and a good source of help, at least in finding where to go.

The final suggestion is library staff. They are generally independent sources of advice.

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    Agreed, I recall very few grad level classes that closely followed a book. One was left to make sense of the lecture notes by taking advantage of the text, the library, etc. – A rural reader Apr 21 at 14:15
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    I like this answer, but just want to expand on the last paragraph. You should also feel free to talk to the lecturer about learning issues. Don't expect them to change their ways based on your discussion, but they may provide context and suggestions on how you can better prepare and learn the material. After all, they have been thinking about how to learn (and how to teach) for their entire educational career (if they are good at it, that is). – Bill Nace Apr 21 at 14:15
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    Out of curiosity, whom should a grad student consult about"learning issues"? University study centers are rarely equipped to help with grad work. – Elizabeth Henning Apr 21 at 14:15
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    Though note that - especially these days - there's often a considerable difference between finding references yourself, and finding GOOD references. E.g. a search for "Chebyshev Polynomials of the first kind" returns some 640K hits. – jamesqf Apr 21 at 15:28
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    I'm not sure if this matters, but in the era of remote learning, there is far less nonverbal communication in lectures. This goes both ways, including lecturers getting a feel for the understanding level of the room. – qwr Apr 21 at 22:21
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I'd like to offer my perspective as a second year mathematics graduate student.

I have found that the courses I learnt the most from were those in which the instructor did not follow the prescribed textbook(s) very closely. In such courses, the instructor would usually spend a large portion of the lecture hours drawing connections and nudging our intuition in the "right" direction. I found this to be very valuable. It is precisely these insights that the instructors offered that made the courses valuable, since I could easily read and follow a textbook line-by-line yet struggle to see the big picture.

Even in the courses where the instructor chose to follow the textbook closely, the standout lectures for me were those in which the instructor went off script and chose to delve a little deeper into the subject, offering their own insights into the theory.

I should add that not all of my classmates share my perception. Some of them did struggle because there was no textbook whose contents matched the lectures close enough, and not knowing exactly which topic will be covered next meant that preparing for a lecture by browsing the material beforehand was not possible. This seems very close to your experience, based on the details in your question.

Without being judgemental, I think this is because my classmates were unused to processing new material on the fly; that is, they were more used to participating in a lecture passively and wrestling with the material privately by reading and re-reading the textbooks or class notes, than wrestling with the material right then and there as the lecture is being delivered. In the former style, the instructor may come into class with well-prepared notes and the students may be expected to jot them down as the instructor delivers the lecture at the blackboard. In the latter style, the instructor may instead choose to develop the theory more organically, often speaking out loud their thoughts and intuitions, often stumbling, and may expect the students to raise questions of their own along similar lines.

Another reason the latter style works for me is that the pace of the lecture becomes slow enough that I can follow comfortably and sometimes also offer my own suggestions for how to get out of a hole. It doesn't work for others because they are unused to being taken for a bumpy ride.

While I don't have many concrete suggestions for how to handle such lectures, I would highly recommend that you try and get used to thinking and responding to the content as it is being delivered. In my experience, such a style of lecturing is closer to how research is done; at least, I see the similarities between the interactions in such lectures with the interactions I have with my guide during our weekly meetings.

One thing that you could try is to form a study group with similar students and discuss the lecture material together. Even try lecturing the same material back to each other. Some of my classmates find it easier to respond actively to a peer lecturing to them as compared to a faculty. The principles remain the same, so I expect this to help, if you can find like-minded classmates.

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    my classmates were unused to processing new material on the fly There's a lot to unpack here. I think you are correct that grad-level mathematics courses often advantage students who do this well, which concomitantly means disadvantaging students who don't. It is not a coincidence that those students are disproportionately from underrepresented groups, which is more complicated issue than "getting used to" it. It's somewhat true that this is how research is done, but I would argue that's a consequence of who gets selected out of this kind of training and not the other way around. – Elizabeth Henning Apr 21 at 18:14
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    @ElizabethHenning I completely agree! I felt that this answer was not entirely the correct place to unpack all the subtleties of that point, so I tried to leave it at that. Even if I understated the difficulties, I hope I did not come across as disparaging. I am acutely aware of how hard some of my classmates have struggled to reach where they are now, especially considering their backgrounds and how it differs from mine. – The Amplitwist Apr 21 at 18:19
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    @ElizabethHenning, you are certainly correct that students who've been led to lack self-confidence can doubt themselves in coping with "live" things. This is one reason I try to emphasize my own fallibility, and the irrelevance of aiming for infallibility, during my "live performances". – paul garrett Apr 21 at 18:21
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Your problem isn't that the professor doesn't follow the text. Your real problem (as clarified in your edit) is an unusual aspect of your question that nobody has addressed. It is your need to "prepare for class" on your own.

In all the courses I can recall, the professor either assigns a specific reading before class, or not. Then the professor designs the lecture to be understood by a student with that specific preparation as the case may be. I've never encountered an expectation that students should guess the lecture content and prepare for it using their own unspecified resources.

It seems that your real problem is that you are struggling to follow class material in real time:

I can only show up to lecture and hope I can keep up.

Your homebrewed solution to this problem is to prepare for lecture by studying the material in advance. So you are asking us how to continue applying this strategy in a new environment. Perhaps you should ask directly about different approaches to this problem instead.

A common answer is that lecture is not guaranteed to make perfect sense the first time. After the lecture, students may need to read or think or review notes, and possibly find additional resources or presentations of the material to understand it better.

But you may have a different learning style than most of your classmates, and you may feel this common approach is not working for you. People on this site may be able to help better if you give more information about why this approach to learning (the usual/expected approach) doesn't work well for you.

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  • It seems that your real problem is that you are struggling to follow class material in real time -- There are numerous reasons a student may want or need to prepare for class (I'm omitting your nasty sneer quotes), including sensory or processing issues, low vision or hearing, or learning disabilities. Expecting that students should follow class material in real time creates a hostile environment for those students who have difficulty doing so. It's also an expectation that has nothing to do with the actual goal of the course, which is learning the material. – Elizabeth Henning Apr 24 at 0:09
  • @ElizabethHenning Aren't there offices in the university tasked with providing accommodations in cases as those you name? I certainly expect students to be able to follow the lecture in real time by default, and anything that interferes with that ought to be dealt with some way or the other (of course I also expect students to take copious notes to be reviewed later and try to facilitate it). If that's an unsafe assumption then the whole concept of university class needs to be reworked... – Denis Nardin Apr 24 at 7:34
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    @ElizabethHenning I apologize if my post came off as nasty, which is far from my intention. You make excellent points and I think that, like mine, they suggest that Square should ask directly about whatever problems they are facing, instead of asking about one solution (X-Y problem). – usul Apr 24 at 13:24
  • @usul Sorry for being prickly about your post. It's always hard to read intentions online. – Elizabeth Henning Apr 24 at 15:03
  • @DenisNardin OK, but what if the accommodation is to provide information (such as a structured syllabus) so that the student can prepare before attending lecture? And why should that have to be a disability accommodation instead of something that is provided by default to all students? – Elizabeth Henning Apr 24 at 15:08
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The typical professor will possess many bits of insight into their field that cannot be found in any textbook. So you are simply asking for the wrong thing. Asking for a professor who teaches only what’s in the textbook makes sense if your goal is to feel like you’ve mastered the topic of the course. If your goal is to actually master the topic at a level a graduate student needs to in order to be prepared for doing research, your best bet is a professor who will curate for you the best knowledge on the topic that they can find, not only from the primary textbook but also from other textbooks and research papers they are familiar with, and from their own personal knowledge.

Do not confuse actual mastery with the illusory feeling of it that you might get by studying in advance the content of a lecture precisely parroting a written source. The latter might feel good and can be acquired in a more systematic and less anxiety-inducing way, but the former is what you should be after.

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If the problem is a lack of repetition of the material, because your lecturers are not repeating the contents of a book verbatim, then you could consider repeating the material by yourself before attending the lecture. Also consider different ways of learning the material; doing exercises, playing around with examples, and playing around with the conditions for theorems to hold.

If anything, I would say it's a good thing that the lectures are teaching you things that are not literally in the book. Be it different but related theory, or just different connections between parts of the book. Instead of 'just' dry material, you are getting some insight into how an expert views and uses the material.

Of course in order to take advantage of such lectures it is very helpful to get familiar with the material before attending the lecture. I can also recommend to let go of the idea that every detail of the lecture should be clear to you before proceeding. Instead, focus on the general ideas and how they relate, and take careful notes so you can figure out the details in your own time. Then you can always ask your classmates, or otherwise your professor, about any details that you aren't able to figure out.

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I certainly grant in advance that tastes vary...

Based on my own student days' attendance in various lectures, and on my own preferences for teaching, I try to think in terms of "added value" that my lectures/classes may provide, beyond any text.

Yes, I do also try to provide my own notes that fill in details that might be tedious to discuss "live". Are more careful about all details.

My lectures, even by Zoom, are meant to be "live performances", rather than reciting notes by myself or anyone else. Live talking and typeset notes are significantly different mediums/media.

I do recall some cases of very nice people, excellent mathematicians, just silently copying their nicely written-out notes onto the blackboard... as opposed to distributing the notes. (Pre-internet, so there were complicating issues...) Worse, to my mind, were the people who thought their job was to recite a chosen text. I could read it myself, at times and at rates chosen by myself, rather than having to show up at random/inconvenient times and sit in uncomfortable chairs ...

Yes, certainly, some improvisational instructors may be a bit irresponsible, in the sense of not providing any written record of their (possibly very nice) lecture material. I myself do aim to provide more-formal typeset PDFs reflecting what I talk about "live".

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I'll guess that this is fairly common, actually. Some things you can do:

Ask a lot of questions during lecture if possible. In a class of 20 or so, it is possible. In particular, ask at the very end of a lecture what will be the topic of the next lecture. This might give you a heads up to look in the text book.

But, and this is actually independent of the question, take a lot of notes. In the notes indicate where you have issues and unanswered questions. Immediately after the lecture, summarize the notes very briefly - just an index card or two. In particular, make a specific note of what the most important ideas are in that lecture.

You can then, annotate those notes with textbook page numbers for later study.

At the start of the next lecture, if possible, ask any questions that you haven't been able to answer yourself from your notes and the book.

For some topics you can use wikipedia, which is pretty reliable for many things (math, for example) and it is organized by topic, not as a course.

If in class questions are impossible, try to find a way to ask questions outside class. Do some work so that you avoid being a bother, but if you have a question, it is very likely that many other students have the same question.

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  • This is the best, most actionable answer, of the bunch. – Julie in Austin Apr 23 at 21:32
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I know it sounds confrontional, but did you ever consider the notion that this class might be too advanced for you? Not everybody is able to deal with a grad course, nor should everyone be.

The fact that you already struggled to deal with the undergrad courses and could only cope by extensive preparation and a very predictable class, might be a good sign that the next step up could be too advanced for you.

It's never bad to discover one's limitations. If you never encounter your limits, you're not trying hard enough. And maybe, with lots of additional time and dedication, you can push your limit beyond what this grade expects of you.

But consider this a sign that you may have reached the limit of your abilities.

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Depending on the technology available in your classroom, a couple of resources that might be available are video recordings of the lecture (so you can go back over the lecture later to make sense of things that weren't immediately clear) or screen capture recordings of things that the lecturer wrote on a stylus and projected on a screen during the lecture (here, you can see what was written but not hear what was said.) The lecturer might also agree to allow you to make an audio recording of the class. If this technology is availabe in the room but your instructor isn't making use of it, then it would be reasonable to ask the instructor to consider using it.

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A perspective of an ex-student, then ex-lecturer and now a parent.

I usually taught math/physics the way I ended up understanding it. I was not a particularly brilliant student so I had to think a lot when learning, trying to keep everything more or less in a coherent state. This meant simplifications, analogies - and then understanding why the simplifications and analogies were only good to a given point.

My students usually found this useful because they had the "wandering around" part and then another approach, usually more formal, in the textbooks.

This is also how I was explaining math and physics to my children: by going with them through a journey to understand (and not only learn) what they were supposed to know.

Only in movies this works fine: they were vigorously protesting because they wanted "just the facts, and not again going through a whole story". I insisted, they protested but eventually gave up. My biggest pride was for them to ask me once to go my way because they understood that they will understand.

This was greatly helped by the fact that French mid- and high-school math textbooks are not particularly good. They love to explain to you what the derivative of a function is by starting with "Given x in the domain R ..." - at which point the children are lost. By discussing with them how to avoid a speed ticket when one sees a "50" road sign it was much easier (in other words: describe a physical context they can relate to, and then walk them through the process of though to get (in that case) to derivatives).

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In my opinion, lecture is not the proper vehicle for providing direct answers or application of formulae – the kinds of things that you'd find in the textbook, and thus a "regurgitation"; rather, this would best be achieved at some recitation or office hours with TAs (who are often extremely helpful in helping students understand course content!). Rather, the purpose of the lecture sessions is to explore important ideas in the subject and link them to profound implications for society, institutions (such as public policy, law, and financial systems), technology, and the natural universe, as well as the human condition. In this way, you may find yourself mired in quite some degree of abstraction, formal proofs, and derivations of central theories, which can seem disorienting without tangible practice; this is normal. Everything may seem strangely uncorrelated or not coalescing correctly in your mind upon first sight, but most of the time there are relevant and meaningful connections that you'll discover after, say, repeated exposure from various places (the lecture, the textbook, the academic literature, slides from other university programs).

As an aside, why should we spend significant moneys or get into unreasonable levels of un-bankruptcy-able debt to hear a lecturer spout the same thing as the textbook? JUST BUY THE TEXTBOOK INSTEAD AND CALL IT A DAY, THEN! (unless your goal is to enter industry, that is, in which case the piece of paper that says you know some such stuff may be prized).

Notably, courses usually indicate how many hours you should be spending outside of lecture by the number of "units" or "course credits" in which they are advertised. Of course, this is only a suggestion. These hours should be spent, ideally, in optimized fashion reinforcing the key takeaways from lecture, and then identifying connections between these and the detailed computations and material particulars contained within the texts. This is the stage in which you should be playing around by making hypothesis about what the fundamental theorems say the output should be given certain use cases and input conditions, including edge cases or otherwise interesting or unusual scenarios, and conducting the analysis to verify that the results match your expectation. If not, what was your faulty reasoning? This practice will prepare you for any upcoming assignment.

As a personal test, determine if you can finish an assignment with little to no notes or other references. After seeing the concept multiple times at this point, it should be easier to compute answers or solve related proofs. It may take time and thought, but shouldn't require the textbook if you understand it "to a reasonable level". You don't need to know everything deeply yet, but rather a moderate comprehension, which is enough at this stage.

In going one step further than expected by the professor, you may locate the most influential publications in the field and other primary source content and read those to achieve an even deeper contextual understanding of the field. They are often free and accessible when using your .edu account, or found in the library. This isn't required, if you have time constraints, but it demonstrates initiative and you learn directly from the originators of the theories in question.

Lastly, there's often a project component, typically towards the end. This is an opportunity for you to see the content in application, and it might be the most important part, allowing you to bring together everything you've learned and use it to build something or conduct an experiment, expressing your creativity in the process. This is when I found that I learned the most, as long as I put forward significant time; if you half-ass it you won't learn anything additional, but if you focus, you may actually come away from the course realizing that you understand the core of that area much more than how well you "know" or "learned" information from a textbook for your previous classes. You'll also find that you have a new-found appreciation for the topic, beyond your initial interest during the first lecture.

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