(1) Students' responses to the prompt in the question, notice, and wonder phase of the inquiry can tell the teacher a lot about their current level of understanding and the sophistication of their mathematical reasoning. It is possible to create a hierarchy of responses. From lowest to highest, the student:

• Finds it difficult to formulate any response.

• Asks for a definition of a word in the prompt.

• Asks for an explanation of a procedure.

• Asserts the truth or falsity of the prompt.

• Shows the case in the prompt to be true or false.

• Notices a pattern.

• Identifies a mathematical structure in the prompt.

• Offers another example based on the structure or a pattern.

• Makes a conjecture from particular cases.

• Generalises for all cases,

• Attempts to give a reason for a generalisation or even a proof.

The teacher should record each student's (or pair of students') response on the board to revisit during the inquiry.

(2) The regulatory card chosen by a student also tells the teacher a great deal about their thinking and independence. The card Inquire with another student might be a sign of anxiety about the nature of the inquiry classroom; a student who selects Decide on the aim of the inquiry wants to set a mathematical agenda.

(3) In conference with an individual student, the teacher can ask the following three questions (from Alan Schoenfeld) to evaluate how well the student is regulating their activity:

• What (exactly) are you doing? (Can you describe it precisely?)

• Why are you doing it? (How does it fit into the solution?)

• How does it help? (What will you do with the outcome when you obtain it?)

(4) Students can self-assess their own development as independent inquirers by using the Learning Journeys designed by Helen Hindle (a head of mathematics teaching in the UK). 

The self-assessment can occur at the beginning, during and at the end of an inquiry to see if there has been progress along the journey.

(1) The teacher can use the Assessment Framework to assess students' activity after the inquiry. The framework is based on three levels of inquiry: structured, guided and open. While the framework includes the use and application of concepts and procedures as part of the inquiry, the teacher could also award levels or grades for curriculum content if required to do so by school authorities.

As students become more experienced, the teacher might introduce the framework earlier in the inquiry to promote mathematical reasoning and inquiry habits of mind. Ultimately, students could be invited to co-construct success criteria. As Kuhlthau and her co-authors say, "The most effective rubrics are those in which students have an opportunity to participate in considering appropriate criteria for evaluating their work" (Guided Inquiry: Learning in the 21st Century, 2007, p. 124). 

(3) The gallery walk involves pairs or groups of students displaying their inquiries around the classroom. Peers visit each display and give critical feedback, which is both supportive and constructive, and suggest future directions for the inquiry.

(4) The teacher can assess students by requiring them to keep a journal. The journal is a record of the inquiry that might utilise various forms of technology and could take the form of a video, voice recording, blog or shared on-line document.

The journal would involve a narrative of the course of the inquiry, mathematical notes made during exploration of the prompt and a formal record of the final outcome (be that a conjecture that could form the basis for further inquiry, an explanation, or a proof). 

The teacher can guide students by commenting on the journal at any stage of its development, something which is easier when the journal is in the form of a shared access on-line document.