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I recently committed to taking a mathematics TA position which begins in a couple of months and entails a weekly exercise class and the correction of problem sets handed in by students.

This is my first teaching experience and admittedly I am rather anxious about it.

My question to those who already have teaching experience is the following:

What piece of advice (with respect to teaching) would you give your younger self before their first teaching experience?

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See also: Mathematics Educators stackexchange. – theindigamer Feb 12 at 16:56
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If you're not boring yourself, you're going too fast. – user37208 Feb 12 at 17:37
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be sure to have good chalk, if the one your department provides is good thats fine, but usually it is not.. that DOES make a difference. more generally: for behaviour in class try to think back about a tutor/lecturer you think did a good job and try to emulate the style until you feel comfortable doing things your way w.r.t hand ins: A) not readable is not handed in B) there is no value in trying to be overly "fair", dont obsess over small details C) never actively try to find the source of an error (if you see it, fine, point it out.. if its work its the students work) – Bort Feb 12 at 18:10
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small addendum (ran out of characters): there is a certain group you are doing this for, interested and at least medium good in the material. you must not solely focus on them, but they/their progress will be the source of your enjoyment. They might be students who are bored by the class because they are to good. Ideally this would be covered by having more advanced problems or time in class to give an outlook into more advanced topics, but this might not be the case. – Bort Feb 12 at 18:14
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I stumbled across the MAA's guide for first-time TAs just a few weeks ago. I can't speak to the quality of information (since I'm a new TA as well), but it certainly seemed to be well-written. – chipbuster Feb 12 at 19:33

16 Answers 16

Make sure you are thoroughly on top of the material before every student interaction.

This sounds obvious. Moreover, it sounds like it could be condescending: if you're, say, a PhD student and you're teaching, say, freshman calculus, haven't you known the material cold for most or all of your adult life? Probably yes...in some sense. But the sense in which you need to know the material cold in order to answer students' questions effectively is much stronger.

Imagine that when you were a calculus student you were assigned to do ten applied optimization problems. Let's say that you knock out eight of them immediately, spending no more than 5-10 minutes on each. There's a ninth one that you keep doing the algebra wrong and it takes you 20 minutes instead, but no problem: you get it. There's a tenth one which has some weird thing inscribed in some other weird thing and the first time you think about it you just get a bit confused. No problem: you put it down for a while, go about your business, and you come back to it later when it's the one thing occupying your attention and you solve it correctly. How good at applied optimization are you? The answer is you're excellent: an A student with room to spare. This level of skill more than qualifies you to be a TA for this material...but you had better do some advance preparation before the session itself. As a TA, the students are very likely to ask you about the two hardest problems. If you make more than one or two algebra mistakes in front of a large group, it's not great for them or for you. If you try to solve a problem and fail, then -- even though you know you'll get it later, and even though "getting it later" is absolutely what matters in the world of actual mathematical work!! -- then the session itself is probably a failure, the course instructor is going to get some "How am I supposed to solve this problem when even the TA couldn't..." questions, and she is perhaps going to be less than thrilled with you. When I was a TA it seemed to me that my on-the-spot problem solving skills actually had to be at a higher level than if I were actually teaching the course: if you are teaching the course, then you are controlling the pace of the information flow and choosing the problems. (In particular, if one of 50 problems from some section of a textbook is slightly confusing to you, you'll probably just not pick that problem!) I haven't had to deal with course TA's for a while -- in my current job, graduate students do grading but rarely TAing -- but upon reflection I still find some truth to that.

(By the way, I have picked what was and maybe still is for me one of the shakiest topics from freshman calculus. Of course there are lot of calculus problems that I really have always known cold...and after many years of TAing and teaching I really know which is which. But there is something in the psychology of math students which makes you think that you should know everything cold or there's something wrong with you, so why don't you proceed under the assumption that there's nothing wrong with you and see how that works out. What I'm saying is that there is likely to be something that you will screw up if you wing it...and that doesn't make you a bad math student, it just means you shouldn't wing it.)

How do you make sure you are thoroughly on top of the material? As a first time TA, I would actually work every problem in advance. Maybe this sounds obvious or most TA's know to do this...but I don't think everyone does. Even as an experienced instructor -- say, at 10 years past my PhD -- I have had the experience of teaching a new course, thinking I knew exactly how something was going to work out, but because I hadn't written it down in as much detail as I was going to cover it in class, encountering unexpected challenges -- more often expository, but sometimes actually mathematical -- when it came to the lecture. In fact this happened to me about two years ago when I taught a first course in linear algebra. I really know linear algebra: like most mathematicians, it comes up in my work, and unlike many mathematicians I have a published paper in the area. But there are certain things that I didn't have to worry about as a research mathematician and did have to worry about as a linear algebra lecturer. I was actually rather embarrassed at the shakiness of my lectures. I compensated for this by being more attentive to the students' concerns and writing more straightforward exams ("How are they supposed to solve problems when even I couldn't..."), and in the end I was surprised that the teaching evaluations were as satisfactory as they were: not great, but no worse than certain calculus classes where my lectures were pulled off exactly as I wanted to but the students were not as thrilled.

By the way, I well know that a lot of people will be tempted to roll their eyes a bit at an answer which is entirely focused on the content of the subject and says nothing about pedagogy, is focused on the mentality of the TA rather than that of the students, and so forth. But this is where you have to start. A first-time TA who knows how to solve all the problems and is just starting to learn about everything else is doing okay...and in fact better than some. The next most important thing is learning to present mathematics in a clear, orderly way and the best and onliest way to do that is to get a lot of practice doing so. The next time you teach the same course is when you really get to concentrate on the "higher aspects" of teaching.

Good luck, and good luck to your students.

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Yes, absolutely work every problem in advance. But if there is a side-question you can't answer off the top of your head, don't worry about it. Just say "I actually don't know right now, let me get back to you about that." and then tell them the answer in the next session or on the course's email list. It's better to admit not knowing something (students can relate to that) than to talk around the answer and confuse everybody. – Sumyrda Feb 13 at 9:42
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@Sumyrda: If you know you won't be able to answer the question on the spot: yes, absolutely, say so and follow up on it later. Don't be afraid to say so: being honest about what you don't know is a critically important teaching skill. – Pete L. Clark Feb 13 at 15:03
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Maybe a more accurate way of saying this is if you see that you'll be stuck on the question or don't know where to start, you should do this. But of course for most math questions that you haven't prepared in advance, you won't literally know the answer off the top of your head...but may know how to get it. I think that being able to field questions not identical with the ones you've prepared, but reasonably close, is part of the preparation process. – Pete L. Clark Feb 13 at 15:03
    
Yes, if you don't know the answer but feel confident that you can get to it and explain as you go, do it. That helps students learn to think like a trained mathematician. – Sumyrda Feb 13 at 23:03
    
I well know that a lot of people will be tempted to roll their eyes a bit at an answer... This is the only answer I read so far that made me shout yes inside my head. – Kimball Feb 14 at 14:57

Some items:

  • Not everyone is going to get it. Like a doctor, you're going to have to accept that some number of people will be irretrievably lost.

  • The median work is so much worse than you ever imagined. As a math-professional, you were probably at the top of your class, getting "A"'s, and following correct mathematical grammar, etc. The majority of work I got when I became a teacher was so shockingly awful I had a hard time processing it. I never would have conceived that was what was going on around me in most of my classes most of the time.

  • Make sure that the correction work is feasible within the dedicated time. If it's taking too much time, discuss that with the appropriate faculty member. Worst case, prioritize and do what you can.

  • When I became an instructor of my own class, my faculty advisor recommended collecting student work notebooks and "checking them off" weekly to confirm work output. This is something that I came to have nightmares about, because the work was so mangled I couldn't even read it or imagine giving credit for total nonsense (i.e., some scribbled BS and then copied answer from the back). I wrestled with that (different iterations or evolutions) for maybe a decade? Finally I just trashed the idea and stopped collecting homework, which saved me a nervous breakdown. That "check off notebooks" idea was one of the worst suggestions I ever got. (Even though I otherwise adored that professor.)

Anyway, that's what comes to mind. Your mileage may vary widely based on your institutional context. Maybe you're at a place where all the students are exceptionally well-prepared, etc., etc.

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"The majority of work I got when I became a teacher was so shockingly awful". My undergrad math professor handed out weekly exercise sheets containing copious practise problems. They weren't for hand in, and we didn't have to do them. He said that in order to pass, we would need to do every single one. I was one of the non-math masses that wouldn't have stood a chance in a rigorous pure math field. I took him seriously and did all of his problems and passed satisfactorily. Many of my peers didn't. I now understand that this was the exact problem he was dealing with. – stacey Feb 13 at 12:44
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I wasn't even a stellar student back in my day. I know something about laziness. But after having to teach, I had the same shock. Due to not being a straight 'A' student I couldn't just lean on my own superiority as explanation (and I think that is a mistake anyway). In practice the reason is often that classes before you have not been understood. Somehow it is suddenly your problem that the stueents didn't pay attention in the trigonometry lectures, that they did not understand programing 101, prerequisities for whatever your doing. Too many pieces of the puzzle are missing. – joojaa Feb 13 at 20:52

One important thing to remember when talking to students in lower-level courses (whether in lectures or in office hours) is to be careful about your vocabulary. Phrases that are used all the time in advanced mathematics, like "necessary and sufficient condition" or even "if and only if" are likely not to be understood. You'll often need to use longer and more explicit formulations than you would if you were talking to people at your own level of mathematical sophistication.

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My advice: After every class, sit down for ten minutes and think about what you think worked and what didn't (at a teacher/student communication level, not what formula you happened to write down wrongly). At times that's a bit painful because you will find that you really tried hard to explain something and the students simply didn't seem to get it. There will also be many cases where you really tried to interact and all you got were sheep's stares.

But over time, if you're honest and critical with yourself, and if you care about improving, you'll find the patterns in the ways you teach, and you'll become a better teacher. You can accelerate this by finding books about teaching college students, of which there are many.

(I like to tell my grad students and postdocs the story that I've come to realize over the years what I'm good at as a teacher and what I'm not good at. This is my 11th year of teaching, and I would say that it took me 7 or 8 years till I really realized that people are different in how they interact with students. I have colleagues who are great in classes of 100 students. They remember everyone's name, they are perfectly prepared down to the commas of every example they want to show, etc. That's not me. I see a herd of sheep in front of me in such circumstances. But I really enjoy -- and think that I'm good at it -- asking my 20 students to move their chairs in a circle and discuss out what a Taylor expansion really is. I wouldn't have done that when I was a young prof, because I had never seen anyone do such things in the classes I had taken. But it's what works for me as a teacher, and I learned these things by introspection and reflecting about what worked in my classes and what didn't.)

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Although very popular in the mathematics community never use the phrase "it's trivial". No one will dare ask a question after that.

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Students can either be thinking or writing, but not both at the same time.

I wish I had known this when I started teaching. I hate to think of the number of times that I put a new slide up and then immediately started talking, or wrote something on the board and talked while I wrote. My rule of thumb is now: "As long as there are more than a handful of pencils moving, my mouth is not."

It takes more than three seconds to decide if you have "...any questions?"

I learned this one when I went to a talk and struggled to keep up with the speaker. She did a wonderful job of constantly pausing to see if there were any questions, but the pause was barely enough to draw breath. If you didn't speak up or raise your hand immediately, the talk went on. On at least two occasions, I was still chewing over her last statement when she asked for questions, and after she had resumed talking, I realized too late that I did have one.

The world trains students to expect immediate feedback. If it's convenient, try to provide it.

There are three blank spaces at the top of all of my gradable work. One is for the grade earned on the assignment, the next is for your current overall class grade, and the last one is your predicted final grade in the class if you maintain your current performance. I use a spreadsheet as my gradebook and when I enter grades, I copy down all three values onto their assignments. It adds less than three seconds per paper to my grading process, and it's been mentioned in student evaluations as being quite helpful.

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I must be missing something obvious, but how do the last two grades differ? It seems to me like if you maintain your current performance, you'll also maintain your current average grade. – Anton Golov Feb 13 at 12:59
    
@AntonGolov - most of the time you'd be exactly right. Current grade is what you would receive if you turned in no further work for the rest of the semester. I include it because it's an actual number. If I only provided a predicted number, someone would certainly come to me at the end of the semester and argue. – Steve V. Feb 13 at 16:06

The best thing you can do is PREPARE. This has been said earlier, but it's so important for a beginning instructor. Write out your examples, mock up your speeches, and anticipate difficulties the night before your classes. After you teach for a while, save some time by mentally reviewing, fully solving only the more subtle ones. Remember that a problem's difficulty lies not only in its execution but its presentation! The effective communication of your work is key.

Speaking of communication...

Boardwork. Think of how your best professors use the board, including their use of whitespace and delineation. It's an art. Like any artform, know your materials: different classrooms have differing setups. Bring fresh markers/chalk with you - using a weak marker can quickly make your lectures feel lukewarm. Make your work vivid: use colors.

Using a board well requires a bit of precognition... You need to predict how a certain concept or example will evolve to effectively use your space. Some information you will want to stay visible through certain examples; other information you will want to make ephemeral, like when you answer a digressing question. You can verbally alert your students to the important or unimportant material. Sometimes you might want them to not copy down notes but simply be attentive.

Your boardwork will be cleaner if you prep, and will hone itself if you reflect. Observing how others' use a board is immeasurably helpful as well; it's like watching an artist paint... poorly or masterfully.

Verbal communication. For the love of teaching, avoid relying on pronouns. Don't solve it - evaluate the line integral from r equals a to b. Don't substitute that into that - substitute the geometric definition of the dot product into the right hand side of the Cauchy-Schwarz inequality. The more accurate you are, the finer you will deliver your point, especially when a student is head down, copying notes. Just think of all those times you wanted to scream your head off when some teacher of yours referred to everything as this: This is vague.

Be sure to explain any technical language, and train your students to speak technically: correct and clarify their questions. It will aid you both.

Repeat yourself. Students are busy assimilating your words with visual cues, scrambling to copy information coherently... They will miss your words and intention, so repeat yourself, pause, and repeat yourself again. Pausing permits processing.

Respect your time. I suggest timing yourself whenever you create course materials and especially when you grade assignments. A lot of your work will become streamlined with experience, but the earlier you focus on your time management, the better. For example, the amount of input you give to your students during grading can consume an enormous chunk of life. Addressing this can amount to a level of restraint. Similarly, office hours should be the students' time, but when office hours end, that means their time ought to end. Give them resources they can access outside class. Remember that they might not yet be strong self-learners, so nudge them in that direction.

Depending on the class, it might be wise to restrict homework questions to office hours and your email strictly to administrative issues. Don't feel uncomfortable setting such rules and boundaries.

And for everyone's sake...

End on time. Otherwise known as respect everyone else's time. Not as easy as it sounds :)

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There are a lot of excellent points here - but I'll toss in a few more.

Looking back on my first semester TA'ing I would give this advice:

  • Prepare an extra 10-15 minutes of lecture.

You'll likely never need to use it on top of your normal plans for the day, but it also helps you look ahead at how concepts might tie together. If you can use one simple explanation and build upon that same example in the future, you're helping students put the ideas together quicker.

  • Thoroughly rehearse before each lecture.

I don't just mean skim notes. Talk outloud if you need to. How will you explain more complicated sections? Do they rely on ideas you assume they know? What are alternative ways to conceptualize this topic?

...and my last piece of advice:

  • Be at class 10-15 minutes early.

This certainly isn't always possible. But, if you can do it, unpack and decompress before you teach. Glance over the topics you want to cover one final time. Let the students filter in early. I connected well with my students, but I also started talking before class began about something: current events, philosophy concepts, logical thinking - often completely unrelated to class. Encourage them to think for their own. Why did they buy laptop A instead of B?

Some days, I just played music I knew they wouldn't normally listen to.

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Another few points to add:

1) Above all try to keep the appointed time. Start on time and also try to end on time. Your TA class is not the only class going on that day and students often have a very tight schedule. Up to 5 Minutes every now and then is fine, but anything more should be postponed to the next class. If it keeps on piling up you should have a word with the professor or skip unnecessary exercises. Or try to schedule an extra class.

2) Make sure to let the students know when the class starts and ends, i.e. have a short greeting at the start like "Good Morning" and also have a very clear ending like "Then we will see again next week".

3) Try to have a very friendly work environment, where students are not afraid to ask questions and feel secure enough to make mistakes in front of everyone. That includes letting other people interrupt you and ask questions on the spot whenever they appear. However also make it clear that constant chatter between students when you or someone else speaks is not ok and it would be better to discuss any questions openly with the whole class.

4) Do not sugar coat when something is wrong, but try to stay constructive for their point of view or their approach to the problem. If I spent a few hours working on a problem and there are still some flaws in my reasoning I really prefer to have these flaws fixed instead of being presented with a completely new approach to the problem.

5) Do not feel threatened by good students that may be even more proficient in the matter than yourself. Make it clear that YOU are in charge of this class, but try to integrate them as good as possible and let them get recognition for it. The whole point of the class is to teach the people in this class and not to show how good you are at this particular topic. If there is someone who can help you accomplish that then feel free to use him/her. Never antagonize or shun them.

6) If there are tough questions you are not able to answer on the spot tell them that and prepare a good answer for next time. Providing half true answers on the spot does not help anyone and will only make you look like a fool later.

7) If you have free time use it to extend the scope of the class. If you have finished all the exercises and still have time left feel free to pose any additional in class exercises and motivate the students to work on it or discuss other problems. Going home early is nice every now and then, but should not be a very common state of affairs.

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Since you have a couple of months to go before you start your duties, you might like to read How to Teach Mathematics by Steven Krantz. This is an ample collection of advice, including, but not limited to voice control, time management, computer use, blackboard technique, course design, motivation, dealing with late submissions and cheatng, mistakes, questions and problematic students.

You might find other advice from this Academia.SE question helpful.

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In particular, there is a free online appendix available here: math.wustl.edu/~sk/teachapps.pdf – Daniel R. Collins Feb 14 at 21:44

We had quite a bit of success by breaking down the sessions into (around) three problems: One solved by the TA (but asking for guidance by the class, not just solved by the TA); one (or a set) solved by students (i.e, give a 3 to 5 student group a piece of the blackboard, or seat them so they can discuss the problem), while the TA is available to help over bumps, solutions are presented/discussed by the whole class; third problem is to be solved individually and handed in, graded on a binary basis (looks reasonable or not, no detailed grading). Average of TA sessions is a 5% of the final grade. All problems and solutions are published after the session. It looks like a short review of the material at the beginning of the session will be required, many skip class or just sleep through it...

The idea is to force students to really go over the material. Otherwise they tend to use the strategy of "bum the night before the exam", with the predictable disastrous results.

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Never think at the blackboard. Coming up with a proof, correction, example, or diagram on the spot is MUCH harder than you think.

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I think you misspelled "Always think at the blackboard." – JeffE Feb 12 at 23:30
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@JeffE I took it more as an admonition to always have your lessons planned out and know the material inside and out, so you never have to think about what you're going to do next. (e.g. walk through a proof at the blackboard using pre-planned steps and explanations; don't attempt to "figure it out as you go along") - But I agree, to be a good answer there needs to be clarification and explanation as to what is meant. – R.M. Feb 12 at 23:58
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Yes, beginners often feel they can improvise calculations or proofs at the board, only to discover that doing this live in front of an audience is much harder than it sounds. Even if you understand the principles perfectly well, improvisation often comes out a little awkward or clumsy, with suboptimal notation and minor oversights. It's just harder to follow. What's worse, people sometimes rationalize it by saying "it's important for students to see how an expert thinks on his/her feet", which is great in master classes but a poor excuse for not preparing your calculus lectures. – Anonymous Mathematician Feb 13 at 0:05
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@JeffE But you need to add the caveat that thinking at the blackboard needs more (much more!) preparation than lecturing from a script. – Christian Clason Feb 13 at 1:20
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I added a sentence to clarify the message. I really do refuse to expand this answer any more than that -- either this simple proverb resonates with your or it doesn't. Extemporizing won't help. – djechlin Feb 13 at 4:12

@yoyostein, most for best vs best for most: When I started teaching, I resolved not to just do the second (to get all students to pass and ignore the great ones); I try to keep reminding myself that you can do the second for 85% of the time, and the first 15% of the time (add one really challenging bonus question to a quiz or hw: on a quiz, this keeps the strong students engaged while the rest are still doing the standard bits). The strongest students are used to being bored in class, so they appreciate it a lot -- and obviously, it is fun to see talented brains at work!

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@djechlin, (not) thinking at the blackboard: When I work with beginning TAs, I typically tell them to over-prepare (as so many have in their answers here), but I put your insight slightly differently: when you begin teaching you'll find that as the distance to the blackboard decreases, your ability to think on your feet also decreases dramatically. We've all seen people stand at the board and have a brain freeze; even experts in the field can have a hard time "seeing" what's right in front of them. This kind of 'performance' issue will get better with time, but yes, don't plan on your brain being functional when you are facing a class..

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This is analysis that I heard from a professor that I find quite insightful. There are basically two approaches to teaching:

1) "Do the most for the best", i.e. Focus on teaching and stretching the ability of the best students in class. Downside: Average students may be lost when you start talking about Galois Cohomology in first year classes.

2) "Do the best for the most". This is the utilitarian approach. Downside: Very strong students will get bored by your teaching.

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I was TA the general mathematics course. What my exercise class made the students interesting wast that I taught the students MAPLE software. Actually it did work and students were completely happy at the end.

Remarks: I started this after 6th session because I was worry that they(IT students) do cheating their complicated for example solving integrals. But I was sure they will forget the mathematical methods after that course except discrete math because they don't need them after that course. Now I am happy that they have Maple in their CVs and if someone ask them general math they can solve it at least bu Maple. Another reason to teach Maple is that it completely make the class interesting and also by Maple we had lots of examples to show several geometrical surfaces to the students which was sometimes hard to illustrate & exemplify.

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I'm not surprised that students were happy that someone else did the work... To turn this into a salient point: Students' happiness at the end of a class is a poor indicator of success (i.e., don't let it get you down if they aren't happy) -- much more relevant is how they feel about your class a few years down the road. – Christian Clason Feb 13 at 1:17
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I honestly don't see what's wrong with this answer. Mr. Ghaffari is not saying that he only taught students using MAPLE or that he did their homework for them. It would be better if he explained in what manner MAPLE was used in the sessions...but I don't really agree with the downvotes. – Pete L. Clark Feb 14 at 20:37

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