I'm an Electrical Engineer, working in industry but planning on returning to school in a few years to get my PhD and pursue academia. I was recently offered an opportunity to be a tutor for high school calculus students, and am curious how beneficial that would be for me in applying to grad school and continuing on eventually as professor. It would only be a part-time thing, but I feel like it would be a good opportunity for me to learn how to teach math, since that's a position I haven't really been in before. I'll likely do it whether it'll look good on an application or not, but I'm just curious to see if it could be doubly beneficial.
I am not an academician and I have little idea as to what to look for when hiring a lecturer, though I had the opportunity to sit through a few of those demonstration lectures from prospective candidates and also go through their feedback forms given to other lecturers.
From this little insight, I believe that any experience is better than no experience. After all, you would be teaching the same students a few years later in college if you become a lecturer.
More importantly, leaving aside what your PhD committee or the hiring committee feels about you it would be a good experience and I feel that you will learn a great deal in teaching. Teaching is an art and you are honing it at a level when it matters the most.
As for leveraging it in your career as an academician, in your future interviews do not just mention that you had experience teaching high school students. Instead, focus on the challenging scenarios which arose while teaching and how you managed it. And then explain how that experience would be useful for you as a lecturer.
A good lecturer has his or own style which is inherited from their senior colleagues and gets mutated through experiences. You are doing good by being proactive and training yourself beforehand.
Finally, if there is a competition for the position of TA in your school, I am sure this experience will benefit you.
Yes - having teaching experience can only be a plus, especially if you are applying to departments that fund many of their students through TAships. As a TA, you will spend more of your time in lower-level courses (such as calculus) than in higher-level ones, so it may be relevant. However, it's unlikely to be a deal-breaker when put up against your research record.
More departments (although I am not in your field) are turning toward putting more value on teaching and outreach compared to decades past, so working in a high school would be seen as very valuable to those departments. Even more so if you are able to work with students who do not typically have access to tutoring, or are likewise underserved.
I don't believe this will be beneficial, and has some minute chance of harming your application. Tutoring of high school students is relatively elementary, and is usually the kind of thing college or graduate students do to earn a little cash on the side. If you're already working in industry, taking on this kind of side gig might indicate that you're not well compensated in your current position, which an admissions committee member might hypothesize is due to performance issues on your part.
If you had a multi-year history of this kind of teaching it might be more beneficial in establishing a record of your love of teaching and interacting with students. If it's a one-off thing of a single semester immediately prior to your application to grad school, it would come across to me as (probably innocent) resume padding.
If you literally have the free time, then I think it's probably fine if you feel strongly about it. But better things to do with that time might be to try and tutor college students (more relevant to your actual teaching duties in grad school), or complete patents or other work products in your current job that could be claimed in a CV.