44
Yes, showing your students that you too can make mistakes, and that real mathematics is not a linear process, are both very good ideas.
However, this does not mean that you should come to the lecture unprepared!
If I seem a bit vehement about this, it's because I've met all too many math lecturers who seem to feel a need to "prove their manlyhood" (or at ...
38
One thing you might do is contrast reading mathematics textbooks with reading novels. I have seen this done at the start of a textbook draft for a course on the Kuratowski closure operators (MESE sketch).
The material is not generally for distribution, but here is a brief excerpt from the start of the book:
I see that Topology was mentioned in a comment ...
34
Students are used to other people being the source of truth.
Even in an algebra class, they will do something (incorrect, at least in the common context) like this:
$(x + 3)^2 = x^2 + 9$
and then ask me if it is correct, or if it actually goes a different way. The implication is that I know the truth and they cannot know it without me. My goal then ...
33
Every student question should be treated like a gift. It gives insight into student's thinking. Even a disrepectful question is an opportunity for you to teach, except that in such a case you don't teach math. There is no such thing as a "only math" teacher. Every teacher teaches life just by standing in front of the students.
I would say it is actually ...
22
Here is something that you will hear from students who attend clear, perfectly-prepared lectures:
"When I'm in class watching you do it, it all makes perfect sense and seems really easy. But then I go home and work on it and I don't know how to do anything."
This is a disastrous outcome in introductory courses because it makes your students think that ...
20
You should avoid a "I am smarter than you are" war with the student, the teacher must be above that. (You're already avoiding it, I am saying so for the benefit of the Internet.) In Japan you could just smile mysteriously, but I suspect we're talking about the Western world.
The first and most important thing to resolve is this: Is your student right? If ...
19
Does this student really already know everything? I ask because of an experience I had many years ago, teaching a "semi-honors" second-semester calculus class in the first (i.e., fall) semester. So the students were there because they had already done some calculus in high school and had done well on a placement test. The first topic of the semester was ...
19
The key is not just that you spend time on a problem - it is normal to struggle with problems at the level you have described - but that you do so in a productive way. In fact, the notion of productive struggle is present in the literature on Mathematics Education; see, for example, my earlier MESE response here.
An interesting source to consult is the ...
18
I don't think it's fair to tell the students to not read ahead. In fact, I'd encourage it. What you should make clear is that the material in the current section takes precedence, and if somebody else has a question on the current material you'll sadly have to help those people before the questions on the next section.
If a weaker student is raising their ...
18
It has been nearly a year now since I've made this question, and I think I've discovered a 'magical cure'. This one simple trick has worked for all of my classes with great success (I am feeling more like a scam-advertisement as I am typing this), though I admit I don't fully understand why it works so well, though I do have theories.
Simply kneel down ...
17
First, make sure they know that:
The purpose of exams is to test students' knowledge and understanding.
The burden of proof is on their side, that is, blank/unreadable sheets work against them.
The teachers might choose to decipher some of the messy work, but this choice is intrinsically unreliable, erratic and may produce unfair results.
The teachers ...
17
I prefer to think of lectures not as the worst way to teach but rather:
A lecture is the worst way to teach, apart from all the other ways we've tried.
That said, to understand when it is appropriate to lecture we have to understand what lectures do. At its heart, a lecture is a time-efficient way of communicating "stuff" from the lecturer to the ...
17
In the first place, it is "only" common course-naming conventions (and the AMS and NSF subject classifications) are to blame for the impression that there is some meaningful schism between something called "pure" and something called "applied" mathematics. Second, as noted, as much as anything people rationalize their own limitations or failings by blaming ...
16
What a good question this is. Others have already given good answers. I think Ilmari Karonen's answer is closest to my own heart. For now let me just respond to one aspect of the question:
I have heard the story (may be an urban legend?) of a top professor who occasionally wanted to teach freshman analysis. He believed in the method of letting ...
16
Consider a paper from this year: Setren, et. al., "Effects of the Flipped Classroom: Evidence from a Randomized Trial", Annenberg Institute at Brown University (2019). In their introduction, the authors write:
Despite the proliferation of the flipped classroom, little
well-identified evidence exists on its impact on student learning.
In this study a ...
15
For reading individual proofs, my colleagues and I have had some success in research studies with self-explanation training adapted for mathematics students. We haven't extended this work to whole sections of books yet, but it was encouraging to find that students could improve their reading after light-touch training - they did better in subsequent proof ...
15
I believe that my students can learn best when they display perseverance (sometimes called grit). I'll discuss how I teach, model, and encourage mathematical perseverance.
Tell a story about perseverance
When I speak to students who are struggling, especially if they failed the class before, I tell them the story of Ella, one of my algebra students. When ...
14
A year after this question was asked, the bloom is definitely off the MOOC rose. The primary finding is that the majority of people who finish one already possess a prior bachelor's degree; offering one to say, at-risk or remedial students has been a failure over and over again. Some links that you should consider:
Recent overview of the field, "The MOOC ...
14
When in doubt, I often decide simply to quote others! A nice choice, in this case, would be someone who started as a pure mathematician, then worked in applied mathematics, and ultimately moved into mathematics education. Luckily, precisely such a person exists in Henry Pollak: Ph.D under Lars Ahlfors, then Director of the Mathematics and Statistics Research ...
13
The first lesson I ever taught was on Standard Deviation to a high school Intro to Engineering class. It went fabulously and I stuck to the lesson plan covering every point on my slide show and getting through every activity only having to shorten the pair and share activity due to time constraints. The students understood everything, responded to discussion ...
13
When I've taught mathematics, getting students to even attempt to read the textbook has always been a challenge. One of the things that I always start with is having students type sentences into this letter scrambler:
http://www.douglastwitchell.com/scrambled_words.php
and see that English text still makes sense if you scramble it up. I then have them ...
13
From Clark, Richard, Paul A. Kirschner, and John Sweller. "Putting students on the path to learning: The case for fully guided instruction." (2012):
Even more disturbing is evidence that when learners are asked to
select between a more-guided or less-guided version of the same
course, less-skilled learners who choose the less-guided approach tend
to ...
13
This is a contentious and highly individual thing, of course, so all answers in this should be taken with a grain of salt. But, if this is your job then:
YES.
Why? First, some practical reasons that apply to any job.
You are not an island. Some topics and/or classes are distasteful to everyone, or at least to everyone who is on the list to teach it. ...
12
I once asked Ted Slaman, after one of his (typically) excellent colloquium talks, what is the secret to giving such a great talk, and his advice to me was: you've got to think like a comedian. What he meant, of course, is not that one is supposed to tell jokes, but rather, that one should explain mathematical ideas with an appreciation for timing and the ...
12
Make clear that the "make sure the proof is correct" is part of the work to be done in the homework. If it is my proof, or yours, or from <famous textbook> that is wrong, the answer is wrong.
(Yes, need to emphasize that even the above cited authorities get it wrong sometimes).
12
By recommending some book you implicitly acknowledge all of its contents (unless specified otherwise). In fact it would be similar if the student used a solution you presented, which happens to be wrong, but you only discovered the flaw when grading the exam. What would you do?
First, I would not penalize the student for copying the solution from the book, ...
12
I would be careful with the type of result for which one needs a lot of new math to digest the explanation.
For example, I would avoid talking about $ 1 + 2+3+.. = -1/12$ because there is basically no way to explain to highschoolers in what sense this could be true. First, one would need a good understanding of the limit and the value of a series (i.e., ...
11
There's already very nice answer of Andrej Bauer, but I would like to view the question from a slightly different perspective. Perhaps one should not call questions silly, but there are questions which we wouldn't want to answer, the main reason usually being that it would not be the best response. To name a few concrete examples:
As a teacher we have a ...
11
Besides mastering the material in the course one thing that the students have to learn during studies is to communicate mathematics in written form. Almost nobody comes to university and is able to write clear proofs or mathematical arguments. You need to communicate that this is part of what they have to learn.
You may grade as harsh as you like if you are ...
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