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1

You mention that these students are fresh out of high school. I'm not an educator but I can tell a little about how I first experienced this. What is hard in the beginning is to translate dense math language to an intuitive picture in your mind. You learn how to do this after some practice but in the beginning this can be daunting. You can guide your ...


17

On the other hand, they are really struggling with injective functions. Even after spending a lot of time, they often say "a function is one-one if every element in the domain has a unique image". Have you asked your students what they mean by "unique" when they say that? The reason I'm asking is that, by some commonly used definitions ...


27

I think you will find that almost everyone has this problem when they first starting to learn rigorous mathematics, and many students will never overcome this difficulty. The following three statements are logically equivalent to each other, but students will almost all find the first to be easier to understand than the second, and will not be mature enough ...


6

In his Abstract Algebra book, John Fraleigh mentions that this is a common mistake beginning students make. Exercise 37 in Section 0 (7th edition) asks the reader to make a pedagogical case for using the terminology "two-to-two" instead of one-to-one. He doesn't expect the new terminology to actually take hold, of course, but just discussing it ...


8

The only example you give of what's going wrong is students' definition using the words "a function is one-one if every element in the domain has a unique image." I suppose it literally translates into a statement that for every x in the domain, there exists a unique y such that f(x)=y. This is really more like a description of the fact that in our ...


0

We could approach defining perpendicularity by the notion of minimizing the distance from a point $P$ to a line $l$. Then we will have an optimization problem with a constraint. If $P=(x_0,y_0)$ and if $l$ is given by $y=mx+b$, then we must minimize $f(x,y)=(x-x_0)^2+(y-y_0)^2$ subject to the constraint $mx-y+b=0$. Setting this up using the formalism of ...


5

Sharing the impressions of a person who earned 2 IMO bronze medals in his youth, but whose dreams of a successful research career were never truly fulfilled :-) Mathematics is, indeed, not only about problem solving, but it isn't only about building theory either. Individual mathematicians may place themselves near one of the end points of this "...


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