What does your schedule look like? I guess it all depends on whether you are in an elementary or middle/ high school. Many elementary library schedules are tied to teacher plans and that is why they are fixed. You may also have a semifix4d schedule if you are at the middle school level.
What can you do about the fixed schedule ? It may work out fine if you are in a small school. And there are advantages to the fixed schedule. A fixed schedule provides the librarian with the fact that he/she will see each class for instruction and every one will have access to getting books and resources from the library.
With the flex schedule, teachers are more able to collaborate with librarians at the time of need. Many librarians with a fixed schedule may not have any time in their day for collaborating with teachers on lessons. So is there any way to get some of that time needed?
Try to work with you administrator who does the scheduling. Think of other ways to accomplish the tasks that you need to have students do during the year.
Book Check
Can you allow book check out at the beginning and end of each day when there are no classes?
Choose and hold: students can put a sticky note on the books that they want with name and teachers name, Then as time allows , library books are checked to the student's and delivered to their classroom , or arrange for a student from each class to pick up.
Self Check out: Teach students to check out their own books. It really is not difficult for most programs .Check you program to see if there is a self check mode.
Instructional Time:
If you have flexible scheduling , you work out with the teachers each week when they will be coming and what they will be doing. You may have a sign up sheet for teacher's who want to just bring students for check out. If you have set times for book check out in a flex schedule be careful to not fill up your schedule so that collaboration time is lost, Here's a question : do they need check out time each and every week? Younger students maybe - older maybe every other week
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Another way to fit in many classes is A/B week. So half the classes meet on week A. The other half on week B. This leaves some times open so there may be times for open library for research or collaboration with teachers,
A variation on this is to have say K-2-4 during the first semester for instruction with 1-3-5 just coming for book check. At the semester change then the 1-3-5 has instruction and the K-2-4 gets a 15 minute book check out time.