First, a direct answer:
Yes, there is work in mathematics education connected to Gadamer. One soure for such studies is articles by Brent Davis; more generally, google scholar yields such connections readily.
Here are some examples of BA Davis' work in which Gadamer is cited:
Davis, B. (1997). Listening for Differences: An Evolving Conception of Mathematics Teaching.
Journal for Research in Mathematics Education, 28(3), pp. 355-376. http://www.jstor.org/stable/749785.
Davis, B. (1995). Why Teach Mathematics? Mathematics Education and Enactivist Theory. For the Learning of Mathematics, 15(2), pp. 2-9. http://www.jstor.org/stable/40248172.
For an article of his with no paywall, check here and, e.g,. foot-note 19 (pdf 11/19, p. 277).
Second, I am not totally clear on what you mean by problem, but here are a couple of remarks about the role of conversation in the teaching and learning of mathematics. (My post, though somewhat long, necessarily elides over heaps and mounds of literature!)
A professor of Mathematics Education once remarked to me that the best problem solvers are those who are good at talking with themselves; this observation can be found more formally in Schoenfeld's (1985) work on Mathematical Problem Solving, in which expert problem solvers distinguished themselves from novice problem solvers by virtue of self-checking / asking themselves if they were on the right track in the problem solving process. For more, see, e.g., my previous MESE response here.
Moreover, features such as accruing mathematical vocabulary and attending to precision when articulating one's (mathematical) thoughts are both consistent with the Common Core State Standards. For example, see the Standards for Mathematical Practice (SMP) and earlier work from NCTM.
Quoting from the immediately aforelinked (emphasis added):
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 corresponding SMP is:
How this manifests in the classroom (or even in a two person conversation) will vary based on the participants. In a healthy mathematics classroom, there should be an ongoing conversation about the new ideas that are coming up. However, engaging in conversation of this nature requires significant background knowledge (both mathematical content and pedagogical) and a fair amount of bravery on the part of the instructor. After all, it is easy to find oneself on the receiving end of a question that is intractable, but not obviously so; it takes a certain deftness to engage with students in mathematical conversation while maintaining the wherewithal needed to move them in the general direction of understanding that a given course intends to develop.
From the perspective of research in mathematics education, one difficulty presented by conversation-based learning is that it is difficult to identify qualitatively how ideas collaboratively emerge. I believe that some of the literature on distributed creativity (e.g., work by RK Sawyer) may be relevant in this respect. On an individual basis, researchers can certainly evaluate student learning before and after units that use discourse-based learning (e.g., using standard pre- and post-tests).
If you are interested in how to implement conversation into the mathematics classroom, then the best recommendation that I have for K6 is the following:
Chapin, S. H., O'Connor, C., O'Connor, M. C., & Anderson, N. C. (2009). Classroom discussions: Using math talk to help students learn, Grades K-6. Math Solutions. Amazon link.
I am confident that these ideas can be implemented in later years, too, and scaled all the way up to the level of research mathematicians conversing with one another.
To close, I might also note that the use of conversation in teaching students mathematics and gaining a better understanding of their (mathematical) understanding can be accomplished to great effect by using clinical interviews. For more on clinical interviews, the best text of which I am aware is:
Ginsburg, H. (1997). Entering the child's mind: The clinical interview in psychological research and practice. Cambridge University Press.
As to case studies on individual students' understandings of mathematics, the classic piece you are likely to encounter is Erlwanger's article on Benny (link without paywall). More recently, I read a lovely write-up about a young child discussing how she knows number names; the citation is:
Brizuela, B. (1997). Inventions and conventions: A story about capital numbers. For the learning of Mathematics, 2-6.
