A kindergarten teacher can introduce ordinal numbers by playing musical chairs. Play music while the children dance or march around the classroom. When the music stops and the children are in their new order, have them call out their ordinal places in line, starting with the student in the first position.
A kindergarten teacher can let her students take turns closing their eyes and picking a large plastic or foam numeral from a tub or bucket. This is best done in groups of 10. When all 10 numbers have been drawn, ask, "Who is first?" and allows the students to decide which number comes first. The student holding the number one comes forward, places his numeral back in the bucket to be drawn by the next group and receives a small prize. This is a good game to play right before snack time, so you can use a snack as a "prize." The process is repeated with "Who is second?" and so forth, until all students and groups of students have had a turn and received a prize.
The ABC online source for teachers recommends an interactive game using two sets of 10 large cards designed to be hung around students' necks. One card set contains the ordinal number words "first" through "tenth," and the other displays the numeric form "1st" through "10th." Once you have distributed the cards to the students, each student hangs a card around the neck of another. Students then seek out their "matching" partners; "first" finds "1st," and so on. After all pairs have found each other, ask the students to line themselves up in proper order.
You can help students recognize and understand the concept of ordinal numbers by asking students to raise their hands when they notice that you or a classmate has used an ordinal number. The students can make a game of seeing who is the first to raise a hand each time an ordinal number is mentioned. Make certain that ordinal number words are prominent in subsequent discussions and instructions, at first pausing very briefly to give students an opportunity to recognize the ordinal number.