Firstly, the fact that you did not sign a contract does not mean you do not have a contract. In general, if the parties have not agreed that the contract shall be made in a certain definite form (Article 166 Civil Code), the parties enter into a contract after reciprocally declaring their concordant intent (Article 153 Civil Code. In other words, a contract can but must not be in written.
Secondly, labor contracts are either fixed-term contracts or non-fixed term contracts (Article 9 Labor Standards Law).
I guess your contract is like other contracts of teachers a one years contract. That means it is fixed for one year.
According to Article 4 Labor Standards Law, you can terminate your contract (both fixed-term and non-fixed term) whithout prior notice, when
an employer misrepresents any fact at the time of his signing of a labor contract in a manner which might mislead his worker and cause him to sustain damage therefrom
an employer, his family member or his agent commits violence or extends gross insults at his worker.
the work specified in the contract has the danger of ruining the health of a worker (…)
an employer, his agent, or other worker has contracted infectious diseases having the danger of contagion.
an employer does not make wage payments according to the contract, or does nor give sufficient work (…)
an employer violates the provisions of a labor contract or labor laws and regulations liable to sustain damage to the rights and interests of a worker.
Under the aforementioned circumstances a worker shall terminate the contract whithin 30 days from the date he becomes aware of the particular situation.
Given that one of that is the case, the law (as far as I know) does not stipulate a notice period for regular fixed-term contracts. But, Article 15 Labor Standards Law says, that in case of a special fixed-term contract for a term of more than three years, the notice must be 30 days.
The same notice period applies if an employer wants to terminate the contract.
As a conclusion, if you give a 30 days notice you should be fine.
Hope it helps a bit