Let's explain what the nine digit Federal Employer Identification Number is for those that don't already know. The FEIN is assigned by the IRS to business entities operating in the United States for the purposes of identification and for employer/employee reporting.
It would be the equivalent of an individual's social security number. But, the FEIN is not considered sensitive information and is commonly distributed by businesses. Also, keep in mind that the FEIN can also be referred to as the Tax Identification Number or TIN.
Back to your question, should you be concerned with giving your FEIN to the customer? Probably not, since it is generally public information that can be obtained by many search databases like FEINSearch.com. But, I think it is still important to understand why the customer wants your FEIN.
If you are selling a service which you technically are as a teacher, the recipient business organization would need your FEIN to issue what is called a FORM 1099-MISC when payments to you are in excess of $600. But, if the recipient as it appears in your case, is an individual attending the class for personal use, then a FORM 1099-MISC would not be necessary. This is why it is important for you to understand your customer's intention in obtaining your FEIN.
I hope this clears up any confusion or worries. Best of luck with the class.