(1) The questions and observations students make in response to the prompt can tell the teacher a lot about 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;

• conjectures about particular cases;

• generalises for all cases; and

• 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 are you doing? 

Why are you doing it? 

How does it help?

(4) Students can self-assess their own development as independent inquirers by using the Learning Journey 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) Inquiries can end with pairs or groups of students giving presentations. The presentations could cover how the students have developed the prompt mathematically or how they organised their inquiry - that is, at the cognitive or metacognitive level or at both levels. Indeed, the most advanced presentation would consider how the inter-relations between the cognitive and metacognitive levels had changed the course of the inquiry.

Public presentations can also be used for peer assessment through questioning or evaluation sheets. Time constraints might make it difficult to afford every student the opportunity to feed back on their inquiry. Perhaps the teacher might call students on a rota.

(2) A reflection sheet is a way to assess the learning of individual students at a mathematical and regulatory level. Here is an example on the sum and product of fractions inquiry. It has been adapted to take account of the regulatory cards the students selected. The sheet is designed to initiate a dialogue between student and teacher.

(3) Asking students to plan the next lesson of the inquiry allows the teacher to assess how far the class (or groups and individuals within the class) has developed the inquiry and how clearly they can foresee the next steps. Requiring a plan also reveals whether students can anticipate the need for new mathematical skills or concepts.

(1) The teacher can use the Assessment Framework (illustrated) to assess students' activity during mathematical inquiry. The framework is based on three levels of inquiry: structured, guided and open. 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). Although the teacher introduces the rubric, she is open to students' suggestions for additions and clarification. While the Inquiry Maths framework evaluates 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.

(2) An alternative to the framework is to use the Assessment Form devised by Emma Morgan (a maths teacher who blogs about using Inquiry Maths here). Emma's form gives a student the opportunity to respond to the teacher's comments. Another idea from Emma is the Guided Poster - a retrospective account of the inquiry (see examples here). The teacher designs the sheet based on the direction the class has taken the inquiry.

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