"Please do not look at your textbook during the test."
では is a construction meaning "in
topic". If the は were absent, テストで would be closer to "at the test (location)".
じゅうじゅつでは~ = In jujitsu ~
日本しゃかいでは~ = In Japanese society ~
もっこうげいでは~ = In woodworking ~
During is also expressed by saying '中（ちゅう）'. Therefore, テスト中は教科書（きょうかしょ）を見ないで下さい。is a possible answer.
I can't successfully complete the first module in Class because of the above sentence. When I write, テストはきょうかしょうを見ないでください。I get the reply that I should use テストで。When I use テストで, the reply says I should use テストは. I've used both the word bank and typing it out with the same result. My answer is alway wrong no matter which way I choose to answer the question.
This sentence wants you to use both particles. テストではきょうかしょを見ないでください
で is used to mark the event location "in the test" and combines with は to mark it as the topic and stress more contrast/add emphasis to the rules of textbook use. (You can't look at the textbook during the test, (but you may be able to in others at other times).
"On the topic of during the test - please don't look at your textbook"