Seeking Advice on Obtaining a Recommendation Letter for PhD Applications

Question

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:

1. 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?

2. 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).

Answer 1

First, since it's not explicitly mentioned in the question, let me quote (most of) the (relatively highly upvoted) comment of Andy Putman from the Mathoverflow post:

...my personal advice as someone who has read a lot of PhD applications is that it is unwise to get letters from anyone other than university faculty who have taught you in some form. There are 3 issues: 1. it makes it look like you don't really understand the grad school application process, 2. people reading the letter will have no way to evaluate this person's credentials, so they won't know how informed their opinions are, 3. this person's main contact with you was before your undergrad degree, so their knowledge is out of date.

As you say in an edit, this person mentored you throughout undergrad so the 3rd issues is not at play. Still, as the person does not know how to write recommendation letters for a PhD program, and what they've mentored you is not particularly advanced mathematics (though good things to know), I think Andy's advice holds for top-tier, competitive graduate programs.

For mid/low-tier programs that have trouble attracting strong applicants, if this person is able to write a good letter showing you're motivated to learn mathematics outside of your classes, you work hard, and you're bright, then I'd say this letter could be helpful. (Though it could also be detrimental if the person writes some things which are strange from our perspective.)

One thing you don't say is what country/type of university you're coming from. If the faculty at your university have PhDs from North America and/or Europe, they should have at least some idea of how to write appropriate letters, and their letters will be most important. But if not, then letters from your regular faculty might not be any better than one from your high school teacher, and so the latter might make more of a difference.

Answer 2

I would recommend, fairly strongly, that you only use a letter from this person if you can't get strong letters from university faculty that you have worked closely with.

That isn't to denigrate this person or your experience, but the fact is that a doctoral program isn't really about "learning math" but about research, and this person says they have no real experience with that. You want letters from people who can honestly and enthusiastically predict your success at math research.

The problem is that most places the number of recommenders is limited (usually about three) and you can probably find people enough who are better qualified for such a prediction.

The other issue is that the competition is fierce and the people who decide are used to more traditional -research focused- sorts of letters.

---

Make sure you find a way to thank this person for contributing to your development.

Answer 3

A strong letter from your secondary school mentor can't hurt, since you did relatively advanced work with him. That he does not have an advanced degree will not matter. But that letter can't be decisive. The PhD admissions committee will want to see recommendations from faculty who know your undergraduate work and can speak to your potential for research.