I am currently in the process of preparing my PhD applications in mathematics, with a particular interest in Additive Combinatorics or Elliptic Curves. As I embark on this journey, I have encountered a predicament regarding the selection of recommenders, and I would greatly appreciate your insights and guidance.
During my 12th grade, I had the privilege of being tutored by a school teacher who possessed exceptional skills in various areas of elementary mathematics, extending well beyond the standard school curriculum. His passion for teaching and his dedication to imparting knowledge to enthusiastic students have had a profound impact on my academic growth. Even after successfully securing admission to a reputable institution for my BS-MS program, I continued to maintain contact with him, as his teaching prowess remained invaluable to me.
Under his guidance, I delved into a multitude of topics that were not covered in my BS-MS curriculum. Specifically, I studied, among other things, everything in books such as Burton's "Elementary Number Theory," Edwin Clark's "Elementary Number Theory," covering topics like Residues, Quadratic Reciprocity Law (with some proofs relying on elementary facts from Group Theory), and even an introduction to Cryptography and the RSA scheme. Additionally, I also studied almost everything from "A Walk Through Combinatorics" by Miklós Bóna, which comprehensively covers Generating Functions, Partitions and Stirling's Numbers, as well as Graph Theory, Ramsey Theory, and certain probabilistic methods. Furthermore, he assisted me in understanding renowned proofs from "Proofs from the Book" that do not require very advanced mathematical knowledge.
Moreover, I am fortunate that my teacher holds me in high regard and truly believes in my abilities. I know that he will provide an exceptional recommendation letter if I approach him. However, I am confronted with a challenge: he is officially a school teacher without a PhD or research experience. While he possesses a wealth of knowledge, his lack of academic credentials in the field of mathematics raises questions about the impact such a recommendation letter would have on my PhD application.
Therefore, I kindly seek your advice on the following queries:
What effect would a recommendation letter from my school teacher have on my PhD application, considering his expertise in elementary mathematics but without a PhD or research experience?
In light of my circumstances, would it be advisable to include a recommendation letter from him in my application?
Thank you for your time.
Edit: A comment by Dave L. Renfro in the Mathematics SE post of the same introduced me to this link and pointed out that I haven't mentioned which country I will be applying for. Although I haven't decided upon any specific choices yet, I will mainly be applying to US, Europe and a few places in Canada.
Edit 2: After some discussion in the comments of the Math SE and Math Overflow posts, I decided to add some clarifications in addition to my original post. I understand that if it was indeed true that my main contact with this person was before my Undergrad degree, then there was no question about it since he has no idea what kind of a student I now am. But, as I have mentioned, he has really taught me a lot of stuff (that weren't in my BS-MS course otherwise) that may even be useful in the areas I will be applying for. Now, I understand that people reading his letter will have no way to evaluate this person's credentials, so they won't know how informed their opinions are. Also, if the teacher has no PhD himself, he almost certainly has not been on the other side of the table evaluating PhD applicants, so it is not so clear that a strong letter by him will necessarily tick all the boxes that advisors (Europe) or PhD programmes (North America) are looking for. Hence, my question (reframed) should probably be whether the negative points outweigh the factors that are in my favour, the main one being that I am guaranteed to receive an outstanding letter from him (to the best of his abilities).