With regard to Math Education literature on proofs: a person to look to is Eric Knuth (Google Scholar).
However, it may be more fruitful to shift from talking about writing (formal) mathematical proofs to discussions around incorporating sense making into the mathematics classroom.
Sue VanHattum comments in another response here that:
Very young children can prove that two odd numbers will always add up to an even number.
For an example of this, consider the excerpt below, which mentions Deborah Ball, and comes from an article by Alan Schoenfeld (citation: Schoenfeld, A. Mathematical Modeling, Sense Making, and the Common Core State Standards. The Journal of Mathematics Education at Teachers College, 4(2).)
In contrast, here’s a sense making example that comes from Deborah Ball’s third grade class. Deborah had a bunch of kids who were playing with numbers—adding, subtracting, doing various things. They noticed that every time they added two odd numbers, they got an even number. One of the kids said, "Is that always going to happen?" She then stopped and said, "But the odd numbers go on forever. We can never test them all, so we can never know." That’s pretty damned good for a third grader.
Another kid says, "Well, I was thinking about seven plus nine, and actually I knew that the sum was going to be even before I added them up. If you look at seven, it’s a bunch of pairs with one left over, and if you look at the nine, it’s a bunch of pairs with one left over (Figure 1). So when you put them together, the pairs are going to stay the same—but the two leftovers become a pair, so everything together is in pairs, so it’s going to be even" (Figure 2). The girl stopped at that point and she said, "But wait! It doesn’t have to be seven and nine. No matter what that first odd number is, it's going to be a bunch of pairs with one left over. The second odd number is also going to be a bunch of pairs with one left over. And when you put them together, you have all the pairs you started with plus the pair you made, so it’s going to be even."
Now that’s a third grader. I guarantee you that every mathematician I know would say that the student produced a completely rigorous mathematical proof. That’s what I call mathematical sense making, and that’s what I’d like to see in our classrooms. The real challenge we face is to support sense making in our classrooms.
The topic of sense making comes up a fair amount in Math Education; it also features prominently in the Common Core State Standards for Mathematics (PDF) as a Standard for Mathematical Practice:
Mathematics | Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
Of course, sense making is something that can (should!) be taught from the earliest years.