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The purpose of this study is to propose a new model for evaluating the students' mathematical understanding. It is very difficult to evaluate mathematical understanding.The reason is that, first, the mathematical understanding have five components, or strands that are interwoven and interdependent. Therefore, the model can be grasped between strands of understanding. Second, from the perspective of looking at the mathematical learning, there is a difficulty to evaluate the understanding. Thus, moving between learning-as-acquisition and learning-as-participation is required. We have developed an evaluation model through the discourse to solve such difficulties. The evaluation model was developed based on Sfard's four properties of mathematical discourse-word use, visual mediators, narrative, and routines- and Peirce's three ascending levels-quality, fact, and law.