I made the corrections I thought were necessary and rephrased some sentences. I'm not a teacher, but have a rich experience in writing these kind of things :) Good luck with the job!
I am writing to you regarding the position of a part time tutor of mathematics enrichment classes after / within school hours.
Starting next month, I will be a PGDE student studying in The University, majoring in mathematics (Secondary) on a full time basis. I have a strong desire to enhance my teaching skills and build up the career
pathin education as soon as possible. I believe the teaching experience in Secondary School will not only help me to train up my teaching and presentation skills, which are essential for a teacher, but also assist me in obtaining thorough understanding onthe material/course program taught in the PGDE course. This post would also enhance my understanding of the general school operation of a schooland relations between units and organizations.
(In this passage)Allow me to introduce myself briefly in. I am a forthcoming student of Postgraduate Diploma in Education (PGDE), majoring in mathematics (secondary) in The University and a graduate from The University in Physics with a second honor (upper class). I have shown my talent in mathematics when I was in secondary school. I have not only obtained nearly straight A’s in all mathematics and physics public examination, but also enrolled in an exclusive program held by The University for the young mathematics talents and finally got an excellent performance prize in this program.
As for my working experience, I have been employed as a project engineer in the company between 2007 and 2009 (if you're still working there, "have been" is appropriate, but if you're not working there anymore, it's better to use the Past Simple - I was employed). The job duty included customer services and facilitating productions in the factory located in China. Those experiences in previous job, gave me a better understanding of solving problems, encountered with students’ and parents’ inquiries and carrying out different projects required by my pane. In the school (if you mean a specific school, put it's name here. If you want to leave it as "school" with no name, you should write "In school" without "the") I was a tutor in a private tutorial center on a part time basis and had experience in teaching
thestudents and preparing the supplementary note. This gave me confidence in handling different cases and learn independently with my previous experience.
In my tertiary education, I am grateful to be chosen as a student majoring in mathematics and physics. Even (when) [I think there's a word missing here, is it "when"?] I switched my major to physics later for my interest on studying what is the role of mathematics in connection with physical reality, the training in theoretical physics gave me many ideas on the usage on mathematics.
In relation to the said above, I truly believe my professional knowledge and experience will provide value helping students improve their mathematics skills. I would (highly) appreciate if you gave me a chance to express my enthusiasm and background directly in an interview.