Database design for school management to keep track of current and previous school years
The Student table is a master table; generally speaking, information about the student is static and would not change. I would recommend adding children tables to track the student activities/matriculation in or out of classes. All this detail data would be inserted or updated as time advances.
From the hip, I'd start with something like this:
/* Student demographic data */
CREATE TABLE Student
(
    StudentID  INT IDENTITY(1, 1) PRIMARY KEY
    ,FirstName VARCHAR(100)
    ,LastName  VARCHAR(100)
    ,DOB       DATE
    ,Address   VARCHAR(150)
    ,/* etc */
);

/* Information on classes */
CREATE TABLE Class
(
    ClassID       INT IDENTITY(1, 1) PRIMARY KEY
    ,AcademicYear VARCHAR(25)
    ,StartDate    DATE
    ,EndDate      DATE
    ,CourseNumber VARCHAR(50)
    ,/* etc */
);

/* What action was taken with the class */
CREATE TABLE ClassActivity
(
    ClassActivityID INT IDENTITY(1, 1) PRIMARY KEY
    ,Activity       VARCHAR(50)  /*Enroll, Drop, etc*/
    ,/* etc */
);
   
/* Classes students interacted with (took action in) */
CREATE TABLE StudentClass
(
    StudentClassID   INT IDENTITY(1, 1) PRIMARY KEY
    ,StudentID       INT
    ,ClassID         INT
    ,ClassActivityID INT
    ,EffectiveDate   DATE
    ,/* etc */
);

/* How students performed in class */
CREATE TABLE StudentClassGrade
(
    StudetnClassGradeID INT IDENTITY(1, 1) PRIMARY KEY
    ,StudentClassID     INT
    ,Grade              VARCHAR(10)
    ,EffectiveDate      DATE
    ,/* etc */
);

It would also be useful to create views to display the student's current/past enrollment as separate queries instead of trying to handle these questions in the data tables. Such as:
dbo.v_StudentCurrentEnrollment
dbo.v_StudentHistoricalEnrollment
dbo.v_StudentCurrentClassLevel

The logic for which class the student is in would be contained in the view, not in the data tables. Structure your tables with solid relationships, then you can ask well formed questions to answer myriad questions.