Over the course of the coronavirus pandemic, the Department for Education has closed schools in England on two occasions. During the second period, starting in January 2021, the minister ordered state schools to teach remotely. At Inquiry Maths, we have argued here that the online environment can be detrimental to inquiry processes in mathematics. Nevertheless, it is important in the current climate that students' education continues to include elements of inquiry. Andrew Blair reports on an online inquiry he ran with his year 7 class:

Our school is situated in an area of high social and economic deprivation in London (UK). Over 70% of students are eligible for free school meals and at the start of the lockdown many did not have devices to engage effectively with online learning. Over half of students on roll have now received Chromebooks from the school. The majority have been donated by charities as the government supply continues to be unreliable.

Online lessons

My year 7 mixed attainment class has 24 students. Online attendance to lessons is high with just one or two absences each lesson. During the first term in school (September to December 2020), we regularly used inquiries, directing the process through the regulatory cards and following lines of inquiry that originated in students' questions and observations. Lessons were full of curiosity and conjectures. Moreover, the class invariably developed a deep understanding of the conceptual field on inquiry.

For remote learning, we work in Google Classroom and each lesson I upload differentiated slides that students can edit. After an introduction from me in a Google Meet, students ask questions by 'unmuting' or through the chat box. They stay in the lesson as they work through the slides. I monitor progress and ask individuals to explain their solutions by 'presenting a window' for the rest of the class to see.

After a couple of weeks of establishing this way of working, I wanted to get back to inquiry. Students were comfortable with the platform and had got used to their new Chromebooks. I presented the number line prompt on a Jamboard and invited students to post notes. Before doing so I considered going over the social and mathematical norms of an inquiry classroom, but I felt we had established such a strong culture of inquiry that I trusted them to use the freedom to post responsibly.

The results are pictured above. I arranged the observations into sections that gave structure not only to their contributions, but also to the inquiry that developed out of them. The sections are:

• What do we notice about the prompt?

• The difference between the products

• Testing with other numbers

• Changing to the prompt

• A new line of inquiry.

Once I had established the structure, it proved difficult, if not impossible, to deviate from it. I had to create and upload the slides before the lessons so the direction of the online inquiry was less responsive to students' interests than in the classroom.

By the end of the first lesson, we had explored cases when the number line goes up in intervals greater than one. By the end of the second, we had completed a table of our results and discussed the patterns we noticed. During that lesson I invited students to listen to an explanation of how we could use an algebraic representation to prove the results. Four students, including Mohammed who had posted a comment about algebra on the Jamboard, joined me.

However, it proved much more difficult to promote the sharing of ideas in the virtual environment. Even though Alim presented and explained his proof that the difference between the products is always eight when the interval is two (above) at the start of the third lesson, fewer students advanced onto an algebraic approach than I expected.

In the classroom, I would have tried to use the interest and excitement generated by the first group to involve more students. Perhaps I would have asked another one of them to explain to the class or for them each to work with individuals. Unfortunately, in lacking that immediate peer support, many students found constructing the proof a step too far.

The inquiry finished with students working on the difference between the products when the numbers are connected in different ways. Although the majority position was that there is no common result for consecutive numbers with the first 'linked' to the second (using Halima's term) and the third to the fourth, Alim and Mohammed both proved the difference is 4n + 6.

This was a strong ending to our inquiry, but I felt we had not tapped the potential of the prompt - or, perhaps it is better to say, not as many students did so as would have done in normal times.

February 2021

Samuel Down, deputy leader of the mathematics department at Durrington High School (Worthing, UK), used the number line inquiry with his year 9 class. The picture shows the students' initial questions. For a first experience with inquiry, it is notable how creative the students have been in proposing changes to the prompt. The question "What if we add the pairs then divide the answers?" always gives an answer of one. The proof that [n + (n + 3)] ÷ [(n + 1) + (n + 3)] equals one could lead the class into more complex proofs.

This year 7 mixed attainment class came up with a rich and varied set of questions and comments. One pair of students asks if the difference will always be two, while another has already started to change the prompt by linking the top two and the bottom two numbers. Other students suggest using consecutive multiples of ten, which leads to the suggestion that the result will be twenty - that is, ten multiplied by the difference when the numbers are consecutive. Already in the initial phase of the inquiry, the class has generated multiple pathways.