Assessment in inquiry (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;
 asserts the truth or falsity of the prompt;
 shows the case in the prompt is true/false;
 notices a structural element of the prompt;
 identifies a pattern;
 offers another example based on the structure or a pattern;
 conjectures about particular cases; and
 generalises for all cases.
The teacher can record each student's (or pair of students') response on the (interactive) board. (2) The regulatory cards students choose also tells the teacher a great deal about how independently they think. A group that chooses the card 'inquire with another student' might be exhibiting an anxiety about the nature of the inquiry classroom; a student who selects 'decide on the aim of the inquiry' aims to set a mathematical agenda. (3) Ask students the following three questions (from Alan Schoenfeld) to evaluate how well they are regulating their activity:
 What are you doing?
 Why are you doing it?
 How does it help?
(4) Students can selfassess their own development as independent inquirers by using the Learning Journey designed by Helen Hindle (an Lead Practitioner teaching in the UK). The selfassessment can occur at the beginning, during and at the end of an inquiry to see if there has been progress along the 'journey'.

Assessment as the end of inquiry (1) Inquiries can end with presentations by pairs or groups of students. 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 interrelations 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. It might be difficult to afford every student the opportunity to feed back on each 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 of two fractions equals their product' inquiry. It has been adapted to take account of the regulatory cards the students selected, which, in this particular case, were predominantly 'ask the teacher / a student to explain' and 'try to find more examples'. 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.

Assessment after inquiry (1) The teacher can use the Assessment Framework (see above) to assess students' activity during mathematical inquiry. The framework is based on four levels of inquiry: verification, structured, guided and open. If your curriculum arranges concepts and procedures in a hierarchical list, then it would also be possible to award levels or grades for the mathematics used and applied during the inquiry. (2) An alternative 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 should design the sheet based on the direction the inquiry has taken. (3) The teacher can assess students by expecting them to keep a journal recording their 'inquiry journey'. The journal is a permanent record of the inquiry and could take the form of a video, voice recording, blog or written document (including a google doc) and might utilise various forms of technology available in the classroom. 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 or a rigorous 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, for example, of a google doc.
Additionally, the teacher might formally assess the journal with a grade or level based on how effectively the inquiry had been regulated and on its mathematical content. This might be carried out in conjunction with a student's selfassessment.
