We elect Jack monitor of our class.
we choose jack as our monitor
这个场合还是用choose吧，elect多用于正式的选举。
We elect Jack as our monitor.
选举一般用elect，elect sb as 选举某人做....
we elect jack monitor of our class.
elect表示选举，推选时，用法是 elect sb +职位，中间不加任何单词。
如：
Bush was elected President.布什被选为总统。
we chose Jack as our monitor
we choose Jack to be our monitor
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