What is the role of a teacher (of mathematics) at university?

Question

Until now I've been completely focussed on research. I did serve as teaching assistant on some occasions, but I must honestly admit that I didn't give it much thought and my only concerns have been:

• being clear and detailed (and slightly slow) in explanations;
• being available for further clarifications;
• being fair (and not particularly harsh) in evaluations.

These are, I guess, the characteristics that I value the most in a teacher. However, now, teaching will be a more prominent part of my career (although not at all the most prominent) and I will be assumed to have a sound "teaching philosophy". So I'd like to ask to experienced faculty (and also to students):

What is the role of a teacher of mathematics at university level?

What are the traits that are most commonly associated to good teachers? Should I read some research papers on pedagogy to find answers the question above and prepare myself to be a better teacher (and in the future also a good advisor and mentor)? Which ones do you suggest?

What research papers on pedagogy can I read in order to develop a "scientifically sound" point of view when it comes to grading, course design, presenting to students, solving exercises in class, and other aspects of a teacher's life?

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I prefer to post this question here rather than on Mathematics Educators Stack Exchange because I really welcome remarks and answers from scholars (and students) working in other fieds too.

Answer 1

Like you describe, I spent my time through to the end of my post-doc position focusing heavily on reseach, and in teaching thinking about the sorts of things you list. I have spent the year-and-a-half since then looking for answers to your question. I don't think I have learned everything I need to yet, but unless you are teaching top students I believe that the things you list are really not sufficient to teach mathematics well at undergraduate level, if you want to achieve anything beyond rote memorization.

One good source I suggest for starting to understand relevant issues is Ideas from mathematics Education by Alcock and Simpson (it's based on UK teaching, but the concepts are pretty applicable anywhere). It helps to describe something of how undergraduate mathematics look to students, which is very different to how you are likely to see it as a research mathematician.

Another book I have been finding helpful is Mathematics Teaching Practice by John Mason. It has lots of specific ideas for different aspects of mathematics teaching, as well as more general theory.

Although education literature for other disciplines can be helpful, I would advise focusing on the mathematics literature first, as many of the things considered good practice in other subjects do not translate well to mathematics.

In terms of what things you need to think about, here are a couple of ideas I've come across so far:

1. definitions:

Students generally do not understand the role of definitions in mathematics because it differs from how definitions function in society. Many will focus on their intuitive idea of an object, thinking this is enough to get by, without noticing when they are likely to fall into traps. However, they are on the whole not very good at building this intuition themselves, so need you to guide them through concrete examples. They are also likely to not identify what aspects of examples are general and which are specific to the case at hand.

Others will work just from the string of mathematical symbols in a definition. This means they don't necessarily have the understanding needed to identify when they have written the definition down wrong.

The students who do best are those who manage to link the intuition with the formal definition and combine the advantages of each.

2. reading:

Initially students are very poor at reading mathematics. They tend to focus on the mathematical symbols, and ignore the logic. They are also prone to thinking that being able to read something is the same as understanding it, and so will often prefer to read the answer rather than work at finding it themselves.

3. the nature of mathematics:

Many students who arrive at university, at least in the UK, are used to mathematics as the practice of applying prescribed algorithms to numbers, resulting in the correct answer. The change in aim at university can come as a(n unwelcome) shock, or can pass them by so that they are trying to achieve something very different to what you are trying to teach them.

4. methods of learning:

What students like best and what helps students learn do not always coincide.

Answer 2

I'm a student. I have always wanted to be a researcher/scientist cum professor all my life. The following are the qualities I would like to be instilled in me (and a few which didn't turn out the way I expected from a teacher) as a teacher at a university level. Note that some of the points are applicable for teachers in general and some specifically for math teachers.

(The following are in no order of preference)

• One of the most important things that a teacher must do is to use the blackboard and more than you use a powerpoint. There is some kind of magic related to using the board for math. One of my professors used to use powerpoints for his classes, and most of the time students (including me) fell asleep.
• If in case you do want or have to use powerpoints, then make sure to create animations like the ones here. One of the fascinating things about those animations is how perfectly it fits to capture a person's attention.
• Encourage students to use online resources like OCW MIT. When I was in my university (graduated this year), neither did I have internet connection, nor was I aware of such a resource online. Encourage and suggest MOOCs from Coursera, edX, etc. which might be relevant to your course. It is not necessary that they should register/complete every single course or resource. Give them two or three options and ask them to pick one.
• Make it an active class. And make it fun too. I learnt a lot from Gilbert Strang and Mehran Sahami. Be enthusiastic in class, that will help students get out of their tiredness. You can also toss out chocolates like Sahami does when students answer/ask questions.
• Solve problems in class - either all types of the same level (easy/medium/hard) or one type with all the levels, going from the easiest to the hardest. The point is to make sure they understand the problem solving process. In addition, solve example problems from the textbook or handout and for every example problem, give them a problem to solve either in class or as an ungraded homework. This makes sure they are actually doing something in a math class rather than just trying to consume the knowledge passively.
• Make sure the subject you're taking can be related to other majors and fields. Most of the technical subjects have some variation of mathematics being used in them. Take Calculus. It is used everywhere, even when we think we don't need it. I always thought it was used only in continuous domains, mostly in physics and not at all in any other fields. I was mistaken. You don't have to make this a mainstream for your class. Maybe as an ungraded homework or just a casual mention when the relevant topic comes up is enough to keep the spark up.
• Go a bit beyond the syllabus. I know this will be hard, but, just a bit will be enough. It may involve deeper concepts, real world applications, etc. That is up to you to choose. Personally, I have always found it beneficial for me to learn a bit more than what is needed so that I won't be caught off guard during exams. This also helps students if they wished to pursue higher levels of the course possibly leading to research.
• Have a review class before exams. Walter Lewin's lectures had reviews for midterm and the finals. It sort of helps to review everything in class rather than spending extra time apart from the newer material to be learnt the day before exams.
• Inform students exactly how many hours they'd need to spend on an avg outside class. I wound up with terrible time management because I had no idea how to allot the required time for each course. This led me to focus on subjects I loved and got better grades and subjects I didn't find interesting for which I didn't spend time - and I flunked in them. Huge mistake.
• Use standard textbooks, especially the ones which have 4 and above ratings on amazon. Ask students if they liked a different textbook, and if it is a better suited one, then switch. Don't use textbooks which you find personally attractive.
• If it is not a problem with your university rules, make sure attendance isn't compulsory. There will be at least one or two students who might find attending lectures boring. To counter that effect, make sure you deliver incredible lectures that makes students feel want to attend rather than have to attend. In my university 80% was mandatory and I found it a huge waste of time for most of my classes.
• Make sure problem sets contain all levels of problems. It helps students to practice properly. This increases confidence. In addition, don't pick problems from the prescribed textbook. Just pick another textbook and give them the problems to solve. For math, I'm sure there are numerous books and resources online.
• Identify the distribution of students in your class. There will be one/two/a few gifted students in your class. Maybe conduct a simple quiz at the end of week 1 or 2 to find out who they are. (I find it hard to explain it in words) As a teacher, you will know who these people are. Now, chances are that they might be bored if the class feels too easy. In this case, add extension problems which are very very challenging. These can be ungraded, but interesting to solve. These gifted students will always go out of their way to solve them. This ensures that their skills don't rust away with time. Another horrible experience of mine.
• Rehearse for every class you take, fully. Practice. If it is a 50 min lecture, give a 50 min lecture to the empty hall. Walter Lewin used to do this for every lecture he gave (he quoted in an interview somewhere on OCW or YouTube, not sure where).

These are the ones off the top of my head. Hopefully, these are enough.

Answer 3

As a student (of courses in mathematics) I really most enjoyed when the teacher had good pedagogical skills - being able to give "Ahaa" moments in their explanations when stuck on a problem and actually "see the connections" between the theory from the lectures and the workbook problems to solve. This can of course be achieved in several ways - drawing good parallels, by showing other well-chosen examples which aid the bottom-line understanding. I have not taken any courses on pedagogy but perhaps try to find out how countries with working educational systems do. Finland is one example of such.

A pedagogue should know and have true interest in optimizing the exchange of knowledge and understanding from lectures/classes to students. I am sure there are tons of material on techniques for memorization/understanding/maintaining or planting seeds of enthusiasm and learning interest/cognitive theories etc. Sometimes the solutions are simple, sometimes not, but let me bring an example:

The brain (especially a tired student's one) needs to be activated to actually learn/understand and create new neuron connections). There are many ways this can be achieved, but keep in mind /why/ I remembered the following: We had a lecturer who, when some really important concept to understand and remember was about to be explained, brought out an arsenal of different tricks. One time he had a bike with the steering wheel connected to the back wheel which everyone who dared could try during the break and which the lecturer himself tried out several times. Yes, people sat up straight, started laughing and they immediately had the focus on the lecturer and the bike. That focus then remained for 10-15min, enough time for him to present the very important concept he had in mind. And I had no trouble remembering or understanding it, because the brain is associative and every event and data which is presented before us while something out-of-the-ordinary happens will have /much/ greater chances to be remembered - that info might be important, so that's why the brain tries to suck it up and absorb like a sponge.

Do not make the classes too monotonic. People cannot keep their concentration for too long anyway (there is much research done at this point too) and it is very enjoyable for the brain to alternate the way it learns. Like, reading theory for a while, then watching a video clip where more is explained and shown in a multi-media fashion, then the teacher may continue on the problem on the board etc.

The bottom line: Just sitting in the same way doing math is great at some times (like when at home in the kitchen with a large cup of tea, with no crowds of people walking about etc so that you can really concentrate for hours) but at the University class where you actually have access to a teacher for help, that time should be used with care to address the ways people learn and avoid the pitfalls which make their brains ignore new incoming information.

Most importantly I think that teachers and lecturers should have a deeper level of communication and both should be well-informed of what was said and done on teh lecture as well as the mathematics class afterwards. This is something I really missed when studying (not too many years ago, got my degree in 2011). It really seemed that way too often "the left hand/right part of brain did not know what the right hand/left part of brain was up to" at the University. Poor intercommunication between lecturers/teachers/students, low level of feedback possibilities from students and student rarely had any non-imaginary possibility to influence the course - it was always like "thanks for your feedback (if any), we will try to address this for the upcoming years.." Yay :/

Another problem that I often scratched my head about was that every lecturer/teacher (especially lecturer) always thought that their course /was the most important course and the only course (worth to concentrate on) for the students. While in fact as a student you probably have three or four simultaneous courses and all are equally important. Basically, attitude. Attitude does a whole lot!

When I had to go to the lecture where I knew that the lecturer enjoyed tormenting the students instead of working /with/ them, I had a completely negative setting in the brain even before the class/lecture...affecting the knowledge exchange very negatively of course.

The above are just feedback, "a student's diary" if you will, from IRL and not anything I have looked up officially or anything like that.