46
It depends on how much time I can afford to spend on the problem. I check, as you did, whether I see a mistake. If not, I try to explain the math in a different way: a different conceptual approach, which is good for everyone anyway; perhaps replace the variables with numerical values, if appropriate; or replace a general function $f$ by a specific ...
21
I have had this problem before with students who always think they're right. If a student continues to insist you made a mistake, when you know that you haven't, then tell the student to hold the thought and ask them to discuss it with you after class. Once the class is over, write the original problem on the board and ask the student to solve it. If ...
20
I agree with @kathleen, that being a young (and female) instructor can lead some students to be less respectful. In this case, we can use that to our advantage...
Long ago, when I was less secure as a teacher, I would be so embarrassed and bothered by making a mistake that it would decrease my ability to teach well for the rest of that class period. I knew ...
19
I'm nearly sure I did this with my child when she was young.
First, establish that she understands that a number, like three, is equal to $1+1+1$. Hold three fingers up and ask her "how many is this"?
Then spread them out and ask the same question. Are we adding $1+1+1$?
Try holding up eight fingers (keep your thumbs down, for example) and ask her to ...
18
I've run into a few students like this. I usually try to convey a few messages.
It is great that you are so interested in foundations and there is absolutely a place in math for people with this perspective. Followed by a recommendation of books suited to their interests: At a variety of levels, I might include Spivak's Calculus, an intro set theory book, ...
17
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 ...
15
You asked for anecdotal evidence.
I was a "gifted student". The school told me to teach myself 11th grade math (Trigonometry and Algebra 2) in 9th grade. I never formally learned algebra 1, but I understood it. They gave me a book for 11th grade math and I learned. I don't think it did me any harm - but I do think I understood the concepts and didn't just ...
15
I have a bit of anecdotal evidence.
I was unfortunately not homeschooled, nor did I have a technical childhood; I spent my childhood painting and writing short stories. I was in gifted classes, but I was not seen as a particularly bright student. Due to bullying I looked for alternatives to the local high schools, and ended up applying to university early ...
14
Taking some extra care to verify the details of a confusing bit of material (preferably with class involvement, intuition-building examples, etc.) is usually an excellent use of class time. Offering to talk about the issue further after class or during office hours is also a reasonable tactic when time is short or when dealing with a particularly insistent ...
13
Your student reminds me of me in my first algebra class, in 8th grade. I insisted that my answer was a 'better' solution to a homework problem. The teacher was an ex-Marine, who took the time to step outside the room with me and say something to the effect of "I understand you disagree with me about this problem, but I'm responsible for teaching this class, ...
13
I don't see my thoughts expressed by anyone here so perhaps I can chip in! I'm currently a graduate student but I was once that kind of bright bored student you are talking about. I had my fair share of boredom, lost motivation, exciting learning, disappointment and enthusiastic crankery. I can tell you that taking more interesting courses and finding good ...
13
The question you are asking has little to do with the particular subject in which the student excels and everything to do with student motivation. The students have not developed the skills needed to study and persevere through difficult classes because everything to this point has come naturally for them. They haven't needed to set time aside for a study ...
12
I think it helps to make it abundantly clear whether or not you would expect an average student in your course to come up with such a "stroke of genius". If you're presenting something that might induce such a "WTF?!" moment, be very up front about it. I even like to put myself on the students' level if it's a particularly ingenious stroke, and will say ...
12
I would discuss with these students the rather healthy point of view of Terry Tao on the subject.
Summary
One can roughly divide mathematical education into three stages: The
"pre-rigorous" stage, in which mathematics is taught in an informal,
intuitive manner, based on examples, fuzzy notions, and hand-waving.
The "rigorous" stage, in which one ...
11
I was a bit like that in my first three years if study, then I read the first volume of Schwartz' Analysis, where he introduces the ZFC axioms. This is the point when I understood that one simply cannot make sense out of set theoretical axioms before having manipulated higher-level mathematics (here "higher-level" is to be understood in the sense of higher-...
11
Apart from quite general issues about smart kids' study skills or lack thereof, my observations over many years gives me the impression that mathematics and computer science offer special hazards/advantages. I'll address mathematics, since most of my experience lies there, but I suspect similar observations apply to computer science (which I've also paid ...
11
The key with students like this is to allow them to dictate the pace of teaching, while providing interesting and challenging material. Now, that is very easy to say, but here are a couple of pointers:
Don't force them to go through the motions with topics they are already comfortable with. Allow them to test out of any unit, and find them extension ...
11
Here are some suggestions for problem sources in English. Some of them are appropriate for very bright students studying geometry or Algebra II, but might nonetheless prove too difficult for students accelerated to this extent.
-Mathematical Circles, Fomin et al.
-Mathematical Problems: An Anthology, Dynkin et al.
-Problems in Elementary Mathematics, ...
11
Here in the U.S. there's been a rise in the last decade of "math circles"; extracurricular math clubs with students of the same age, with some amount of play/competitiveness to hone their interest.
Disadvantages: They may not solve the problem of being bored in regular class; they may be expensive (not available to the economically disadvantaged); and in ...
10
I've mentored roughly a dozen year-long undergraduate senior research projects, and I've always used a mix of the following techiques to keep students motivated.
Set clear goals, both short and long term.
Students often flounder when they don't understand quite what they should be doing. Research is hard to figure out, and students often don't know how to ...
10
I use a technique I learned from my pilates teacher: in class activities, have different options for different ability levels.
I have worksheets in almost all of my lessons. I almost never cover all the problems in class (solutions are published later). Some of the problems are there to challenge those who breeze through the easy problems, while I can ...
10
To supplement the other U.S.-centric answer: yes, the U.S. standard curriculum through high school is not now and has not been in recent years comparable to several western European or former-easter-bloc education, for a variety of reasons which are irrelevant to the question. But, either way, there are some implicit hypotheses that are (in my opinion) worth ...
10
My immediate response is 'wait a few years'. I've spent a fair amount of time with 3 year olds, and most of them are busy learning how to be a person in their own right, how to have a conversation, what the difference is between real and make-believe, and (often) how to tell when they need the toilet. I've read that they can't understand metaphors by that ...
8
Solutions that could not have been concieved by the average student
It is important to realise that many more solutions can't be concieved by the average student than we usually realise. Even solutions we think are quite routine can seem like strokes of genius to students new to the area.
Students often say to me, "I can follow the lecturer's proof/...
8
My experience is that students that are ahead, and most importantly, asking questions too - are more likely to be in the position to be both independent learners, and able to take on peer-to-peer tasks related to teaching and learning.
This leads to a few options which might include:
Use the advanced students to help the slower students too, thus ...
7
Humbly accept that you could be wrong, and be open to feedback. Turn it into a teaching moment when possible by inviting the student to come up and show their solution. If they are wrong, don't tell them why, but instead ask they class if they see any problem with the approach. It will not only get the class involved, but it will humble the student from ...
7
In defense of enthusiastic crankery, I feel it's very valuable to try to define "infinity" as a number. You can ask the student critical questions, like how to deal with the "$\infty=\infty+1\Rightarrow 0=1$" dilemma. Let him resolve the apparent contradictions on his own; he'll either conclude that infinity isn't a number, or make up some other algebraic ...
7
As per the comments above, I'm going to tackle this explicitly for discrete math and set theory being a shared course. What follows is some strategies I've tried and how well they worked (or didn't work!).
Group Work. Asking the students to do group work (either in class or on the homework), and assigning groups can really help this issue. Imagine that ...
7
Yes, I think most arguments are "routine" in the sense of being very similar to others in the relevant context, while now-and-then there is something "out of the blue".
The latter can be deconstructed a bit, by admitting that all the goofy attempts that failed were not shown, but only the one that succeeded. Thus, possibly, if all the other attempts were ...
7
When I taught high school, I offered each students a choice between doing two different worksheets, making it clear that one worksheet had many problems at a low level for lots of practice and only a few challenge problems, and that the other worksheet had fewer repetitive problems and many challenge problems. My students generally self-selected the ...
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