教学 means teaching, you can use this word in scenarios like classroom, universtity, school etc. Its literal translation is "teaching and learning", and that is why you can not use this word like this: " 教学我/ 教学学生". It is wrong, because you can not say "teaching and learning student". This word is used as a noun or adj. for instance, you want to say "his teaching is great/ his teaching method is great", and that is "他的教学很棒" "他的教学方法很棒" in Chinese. 教导 is not usually used in scenarios like classroom, schools etc, (although it is translated as "teach" in English), it is more like "having an influence on", or "nurture". Thus anyone can 教导 you, like your dad or an old lady, not just teachers. When you want to say your philosophy/ thoughts/ values are influenced by someone, this is exactly where you should use this word. eg: "My dad taught me money is not everything." "我父亲教导我金钱不是一切." "Karl Max taught us communism is the ultimate social form." "马克思教导我们共产主义是社会的终极形态." 教导 is usually used as a transitive verb.

教学 is a noun, it does NOT mean to teach, it means "teaching" as well as education or course. 教导 can be used as a noun or a verb and it means to guide, to advise or guidelines and advice when used as noun. Normally 教导 is used in situations where someone has made a serious mistake, such as committing a crime, it is also used when trying to guide/help someone who doesn't conform to a norm wildly accepted by the public. use this word with caution, it might make people think u r impolite.

I think 教学 means my teacher tell me how to learn and what to learn, 教导 means someone tell me what is right or wrong, like some philosophy of life.

In a short and easy way: 教导=instruct or instruction 教学=teach But they are almost the same.

教学means to teach,its meaning is pure and simple,but 教导＝教学＋引导sometime it means to lead or to guide,for example,when a student need to learn chinese,some one would like to teach him,when a student made a mistake,there is maybe a teacher would like to guide him to the right way or correct him,besides that,there is another deference,教学 is a intransitive verb，you can say 教我 & 教导我 but not 教学我。