When writing out a date, for example Monday June Thirteenth, does the number 13 need to be capitalized or can a lower case "t" be used? Writing it with a lower case "t" some how looks improper but I can not find any rules telling me either way is incorrect. Monday June thirteenth.

Any help, explanation, or rule, would be greatly appreciated. I teach an ESL class and one Japanese English Teacher insisted that the number had to be written with the lower case "t" and writing with the capitalized "T" was incorrect.

The main problem is that we do not normally write out the numbers. One place it does appear is on formal invitations, where it might indeed be capitalized, depending on the overall format of the invitation card. However, it would 'normally' be in lower case otherwise.
Thank you for your prompt response. Can you guide me to any location that would give me a solid reason or answer for this problem? The JTE will not take my word for it alone, and most grammar guidline books like Mike Swan's Practical English Usage 3rd Edition do not capitalize numbers when being written, however, another teacher stated that since the words in the date stand independent of each other within the date format of, as in the example, Monday June Thirteenth, they need to be capitalized. Does that make sense?
Students: We have free audio pronunciation exercises.
I'm afraid that I cannot point you anywhere in particular, but I'm sure that all grammar books tell you that the names of months and days are capitalized, while ordinal numbers are not mentioned. What the other teacher says does not make sense to me. How 'independent'? Like '100 degrees Celsius'?