We can break down the problem into four parts corresponding to each role that needs to be filled:

1. Choosing the President: There are 4 sitting presidents (one from each class), so we have 4 options for choosing the president.

2. Choosing the Vice President: Similarly, there are 4 sitting vice presidents, and we choose one of them. Thus, we have 4 options for the vice president.

3. Choosing the Secretary: There are 4 sitting secretary-treasurers, and we choose one of them, giving us 4 options for the secretary.

4. Choosing the Treasurer: The treasurer can be chosen from anyone who is left after the previous three selections. So, we cannot pick the president, the vice president, or the secretary. Since each class has 3 representatives (president, vice president, secretary-treasurer), we initially had $4 \times 3 = 12$ eligible members. After selecting the president, vice president, and secretary, 3 people have been chosen, leaving $12 - 3 = 9$ people. Thus, there are 9 remaining choices for the treasurer.

Total Number of Ways:

To find the total number of ways to select all four positions, we multiply the number of choices for each position: $4 \times 4 \times 4 \times 9 = 576$

Therefore, there are 576 ways to choose the president, vice president, secretary, and treasurer for the all-school council.

---

Would you like any further details or clarification on this?

Here are 5 related questions to explore:

1. How would the answer change if two roles were required to come from the same class?
2. What if the treasurer could only be chosen from sitting secretary-treasurers?
3. How does the outcome differ if the treasurer must be from the same class as the president?
4. How would the number of choices change if there were 5 classes instead of 4?
5. What if all four roles had to be chosen from a single class?

Tip: When calculating combinations, breaking the problem into smaller parts and focusing on restrictions for each choice is often an effective strategy!