A linear syntax for natural deduction proofs in first-order intuitionistic logic is presented, which has been an effective tool for teaching logic. The proof checking algorithm is also given, which is the core of the tutorial proof checker Tutch. This syntax is then extended to proofs on the assertion level which resemble single inferences one would make in a rigorous proof. The resulting language has only four constructs. Checking of these proofs is decidable, and an efficient algorithm is given.