If a teacher/rebbi scolded a student for making fun of another student, and this rebuke was given in front of others, which embarrassed the student, does the rebbi have to ask that student for forgiveness? If not, why not?
Rambam, laws of Torah Teaching, Chapter 4:
ד,ז [ה] לא יהיה התלמיד בוש מחבריו שלמדו מפעם ראשונה או שנייה, והוא לא למד אלא אחר כמה פעמים: שאם נתבייש מדבר זה--נמצא נכנס ויוצא לבית המדרש, והוא אינו למד כלום. לפיכך אמרו חכמים הראשונים, אין הביישן למד, ולא הקפדן מלמד.
ד,ח במה דברים אמורים, בזמן שלא הבינו התלמידים הדבר מפני עומקו, או מפני דעתן שהיא קצרה; אבל אם ניכר לרב שהן מתרשלין בדברי תורה ומתרפין עליהן, ולפיכך לא הבינו--חייב לרגוז עליהן ולהכלימן בדברים, כדי לחדדן; ובעניין זה אמרו חכמים, זרוק מרה בתלמידים.
A student shouldn't feel embarrassed if his colleagues understand something after the first or second explanation, but he needs more -- as if he's too embarrassed (to ask), he will have come and gone from the study hall without learning anything! Therefore our Sages said, "someone too embarrassed can't learn, and someone too demanding can't teach."
However, this only applies if the students don't understand because the subject is too deep or their intellectual capacity too narrow. But if the teacher feels that the students are goofing off and therefore not understanding, he is obligated to act agitated and verbally humiliate them, in order to sharpen them. And in this context, our Sages said, "throw bile at your students!"
So in theory -- I repeat -- in theory -- there can be instances where a teacher has the right to embarrass a student. When is this appropriate, when is this necessary, and will it have the desired effect -- those are much more difficult questions. So perhaps the teacher might apologize "I'm sorry I went too far", or "I'm sorry if I misjudged and felt this was the only appropriate way, when there may have been other options." But he wouldn't say "Halacha forbids me to ever, ever embarrass a student."
See also the conversation from Nachum Lamm and Joel Rich on TorahMusings, about Rabbi Joseph Dov Soloveichik:
a story of the time the Rav humiliated a student in front of the whole class for asking a question he felt was not so good. Of course, the Rav apologized (after, it is stressed, we are told that he realized the student’s question was a good one, leaving unsaid the question of what if it hadn’t been). ...
I remember reading that The Rav was raised in learning this way by his father, perhaps due to the fact that such geniuses could only understand a poor [logical argument] as a reflection of intellectual laziness. It wasn’t until the year of triple aveilut [1967, in which he lost his mother, brother, and wife] that he realized this wasn’t true.
It is important to note the famous משנה in פרקי אבות ד:יב that writes; "יהי כבוד תלמידך חביב עליך כשלך" – “The honour of your student should be as dear to you as your own”. Therefore, in this context, it would be fair to say that in the same way an adult would feel upset upon being embarrassed and would appreciate an apology the same would ring true for a child.
This scenario brings to mind a famous מעשה that happened with the Steipler זצ"ל:
It once happened that he scolded a child mistakenly. When he realised that he had made an error, he wanted to seek the child out to ask for מחילה. However, the Steipler realised that קטנים לאו בני מחילה נינהו, children under the age of בר מצוה cannot forgive. As a result, he kept this matter stored aside in his heart. When the boy turned thirteen and celebrated his בר מצוה, the guests at the סעודת מצוה were surprised by an impromptu visit of the גדול הדור, none other than the Steipler himself! As one would expect, the Rov was immediately ushered to the top table to sit alongside the בר מצוה boy. Using this as his opportunity he was able to have a few moments to speak with the boy in confidence. The Steipler departed shortly afterwards. The boy would later reveal that the Steipler had come to ask for forgiveness and to accord the בר מצוה boy some כבוד by sharing in his שמחה.
Moreover, One must be cognisant of the fact that בין אדם לחבירו is not just between adults; any resulting prohibitions also apply equally to children. The תורה famously writes, "הוכח תוכיח את עמיתך ולא תשא עליו חטא" – “You shall surely rebuke your friend, and do not bear a sin because of him” (Vayikro 19:17). רש"י and ספרא understand the verse to mean that although a person is required to admonish wrongdoers it must not come through embarrassment. This notion gives rise to the negative מצוה of מלבין פני חבירו – publicly humiliating a person (see ספר החינוך מצוה רמ and סמ"ג לאוין ו). Such an act is regarded as a terrible transgression and is something that חז"ל comment, results in no share of עולם הבא (See בבא מציעא נט as well as משנת תורה, הלכות תשובה, ג:יד). This is because the act of publicly shaming a person is like killing them, and it is actually better to give up one’s life first, than to cause public embarrassment to a fellow Jew. (מסכת בבא מציעא נט – The text reads; "נוח לו לאדם שיפיל עצמו לכבשן האש ואל ילבין פני חבירו ברבים" – “Better had a man thrown himself into a fiery furnace than embarrass his friend publicly.”)
Rav Pam asserts:
"ואין היתר להורים ומורים יותר משאר בני אדם בענין זה אלא אם הוא לצורך חינוך ומוסר לטובת הקטן אבל שכיח טובא שהנזק בזה מרובה על התועלת" “And there is no more permission for parents or teachers [to cause a child to feel shame] than for anyone else, unless it is for the purpose of education or teaching for the good of the child. However, it is far more frequent that the damage caused by this is greater than the benefit.”