misunderstandings due to prejudice

I'm not sure that prejudice is quite the right term here. What I mean is an idea that you have before
you are told about something and that you think applies to that something without justification.
It then creates havoc, especially if you are not aware that this "assumption" is being made unwanted.

You might be making this mistake because you mistakenly think the person trying to explain something to you is really trying to explain something else.

This kind of mistake occurs quite frequently in older people with so much experience, they think they have seen it all and dont listen carefully to what is being said.

A misunderstanding I had in calculus

Preamble:
  I learned basic calculus at school, but curiously never got back to it (apart from basic measure theory) at university where this was considered as being part of "applied mathematics". Certainly we never did multivariate calculus at univerisity.

Since this keyboard does not allow me ro write integration symbols and since I am a bit too lazy to draw a picture I'm going to represent it by the word "integrate".

At school we had always studies things like
integrate f(x) dx
Sure we had been give meaning of this but with time I began thinking of this as "find the primitive of f".
I then got the idea that the d in dx was rather pointless, it was just to indicate which of the parameters was the variable.

But when I got to advanced calculus on my own this way of thinking really prevented me from understanding expressions like

integrate f(x).ds
(for a line integral, we've left out the curve parameter)
where the dot above  is supposed to represent the dot product.

Now this time I was pretty stumped for several reasons:
1. Here s is not a variable to be found anywhere in f(x)
2. The dot product with ds really invalidates my previous approach as to the role of ds.

I must point out that the notation using dot ds is not perfect, because s is not a variable name that can be replaced by another one, it has a specific meaning of an infinitessimal tangent vector and this has to be recognised!

Interpretation and understading of text

The success of undestanding a statement (in its context) is often a case of being find the right interpretation of the statement, but certainly this is not the only problem.

This is very easy to overlook when a teacher or document writer is writing a text:

the writer does not always notice that the text can be interpreted in many ways.

The above statement is not world shattering but we are trying to get a pretty comprehensive list of causes of lack of understanding.

Can you do too many exercises?

I had a look at some university undergraduate mathematics textbooks on subjects I would need to do some catching up. I notice that there are a huge number of exercises. Now the question comes up in my mind:
if I do all those exercises will I be all that better off?
Somehow I don't feel so. I often feel that if you drown in exercises you
1) can lose the feeling that there might be more advanced topics
2) cant' see the forest for the trees.
I'm not saying that I'm certain of these assumptions, but it's worth thinking about.
It's interesting to compare some self-contained advanced texts like Walter Rudin's Real and Complex analysis
with more elementary texts. Of course one might argue that you should read the text that corresponds to
your level of maturity. This answer is a bit pat.
Rudin's book is known for very short clever proofs of theorems in analysis. At the same time it could be argued that  slick proofs might not help understanding the subject as much as stodgy proofs.

Introduction

The point of this blog is to get to grips with the idea of "understanding".
My background is science and mathematics with an interest in cognitive science and artificial intelligence.
I've always dreamt of having a program that "understands explanations", but that wont be the main focus here. Instead I'm mainly motivated by what makes good teaching.
I start off with some general questions which are not supposed to be taken as entirely meaningful, just entries into discussion, as is the case in philosophical discussions.
In fact I asked a philosopher friend about what philosophers had to say about understanding and got the impression he did not want to deal with the question, but conceded it was interesting.

I want readers to participate with comments or even start their own blog and tell me.

I'm not expecting the notion of understanding to be well defined, it could cover a constellation of concepts.

I'll be looking at understanding in different fields like mathematics and physics as well as hopefully some wildly different fields.

I'll be interested in what are impediments to understanding, even if they seem "off topic". For example what is called by some as "math anxiety" is a fear that can be an impediment to anxiety.

And then there is the phenomenon that sometimes you understand a subject as a student, but several years later you can no longer understand the same books. Why? What can be done about it?

The posts will be rather rambling and freewheeling in nature, so bear with me, and please intervene constructively.