I am currently working on a project investigating why students tend to struggle when they first encounter Real Analysis and what can be done to improve the situation. I would be very grateful if any of you could offer your perspectives on this matter and/or point me in the direction of any literature/sources that focus on the teaching of Real Analysis (or Maths in general).
Many Thanks.
Teaching real-analysis is a really difficult game. Professionally I am not a teacher but I have interacted with many students of mathematics and I have found the basic problem faced by most of the students (even some with high grades too) is that they try to use the algebraic approach of symbol manipulation to solve problems in real-analysis.
To elaborate a bit further most of the other topics in mathematics deal with equalities whereas real-analysis deals with inequalities at the most fundamental level (this is seen in definition of limits).
Ask a student to solve $x + 5 = -2$ and almost every student of age 12 would be able to solve it instantly and get $x = -7$. Ask the same student to solve $|x + 3| < 4$ and most probably he will take significantly longer time to find the answer as $-7 < x < 1$. This is what I consider the gap between algebraical thinking and analytical thinking (thinking needed to deal with real-analysis).
Another slightly harder example is the following.
1) Show that there is no rational number whose square is equal to $2$.
2) If $a$ is a positive rational with $a^{2} < 2$ then show that there is another positive rational number $b$ such that $a^{2} < b^{2} < 2$.
The first statement above belongs to algebra and the second one belongs to analysis. As an educator of real-analyis, a teacher must ensure that his students can prove $(1)$ and are equally good at proving statements like $(2)$.
How to enable students to make such a leap from proving statement $(1)$ to proving statement $(2)$? They must be told that there is more to maths than calculation/simplification/computation. And this has to be done from young age of say 14-15 years. Students need to think of numbers as representing magnitudes and appreciate the inequalities between numbers as representing concept of big/small magnitudes. A great help is offered by visualizing inequalities on the number line.
Next step is to give them a serious conception of the "infinite" and emphasize on the statement that there are an infinity of natural numbers (integers as well) and there is no smallest positive rational number. While mentioning these concepts avoid any use of symbols and try to explain the concept in the "medium of instruction" (say English). Then introduce the real numbers properly and without any axioms. Like the way a student learns that a rational number is constructed out of integers via algebraic notion of "ratio", the student has every right to know how reals are made out of rationals via the non-algebraic notion of "inequalities". The best approach is to use Dedekind Cuts and again no symbolism please.
Thus prefer the following
"Divide all rational numbers into two sets $A$ and $B$ such that every set is non-empty and each member of $A$ is less than each member of $B$ and call this pair of sets $(A, B)$ as a Dedekind Cut"
compared to the following
"A Dedekind Cut is a non empty set $A \subset \mathbb{Q}$ such that $x \in A, y < x \Rightarrow y \in A$."
Once the real numbers are introduced properly the machinery of calculus/analysis can be developed rigorously.
I think most of the problems in teaching real-analysis are due to modern trend of avoiding development of real numbers (especially popularized by Rudin who considered that "it is pedagogically unsound to teach construction of real numbers before teaching machinery of analysis").
One more point which is important is regarding symbolism. A student needs to told (perhaps many times) that Greek letters like $\epsilon,\delta$ are to be treated at par with Roman letters $a, b, c$ and their presence in analysis is historical. It would have been much better if $d, e$ were used in place of $\delta, \epsilon$.
