6. An apparatus comprising a processor and a memory storing instructions that, in response to execution of the instructions by the processor, cause the apparatus to at least: receive written input, the written input being representative of a mathematical expression; analyze the written input to identify at least one operator and at least one operand; construct an expression tree based at least in part on predefined symbol relationships, the at least one operator, and the at least one operand; wherein construction the expression tree includes inserting, based at least in part on the predefined symbol relationships, a placeholder within the expression tree for a missing operator, a missing operand, or a missing parenthesis and wherein the placeholder separately identifies whether the placeholder is for a missing operator, a missing operand, or a missing parenthesis; analyze the expression tree to identify an error based at least in part on one or more predefined rules; and provide for display of a feedback message corresponding to the error, wherein the feedback message may include a suggestion for correcting the error.
6. An apparatus comprising a processor and a memory storing instructions that, in response to execution of the instructions by the processor, cause the apparatus to at least: receive written input, the written input being representative of a mathematical expression; analyze the written input to identify at least one operator and at least one operand; construct an expression tree based at least in part on predefined symbol relationships, the at least one operator, and the at least one operand; wherein construction the expression tree includes inserting, based at least in part on the predefined symbol relationships, a placeholder within the expression tree for a missing operator, a missing operand, or a missing parenthesis and wherein the placeholder separately identifies whether the placeholder is for a missing operator, a missing operand, or a missing parenthesis; analyze the expression tree to identify an error based at least in part on one or more predefined rules; and provide for display of a feedback message corresponding to the error, wherein the feedback message may include a suggestion for correcting the error. 8. The apparatus of claim 6 , wherein the apparatus caused to construct the expression tree includes being caused to construct the expression tree as a self-contained complete binary tree.