Helen Hindle devised the prompt to generate short inquiries with her lower attaining classes in years 7 and 10. The year 7 class used the question and comment stems to express their uncertainty about the "little number". These are their questions and observations:  Why is the 2 on top of the numbers?
 I know the little 2 means 'squared', but I don't know how to square.
 This is a sum. I think it is true because 2 x 5 = 10. At the moment I don't know what 2^{2} or 5^{2} mean.
 2 x 5 = 10. With the little 2, it still equals 10. I don't think it will make a difference to the numbers.
 I think it is false because 2^{2} = 4, 5^{2} = 10 and 10^{2} = 20.
 I think the real answer is 14.
 I think it is false because 2 x 2 = 4 and 5 x 2 = 10. So it should be 40.
 5^{2} = 25
Contrast these responses to the initial comments from the year 10 students who spontaneously suggested changes to the prompt and also attempted to generalise by considering other cases. In the top righthand corner of the board we can see three students speculating about the law of indices: one claims 2^{2} x 3^{3} = 6^{6} because you multiply the indices; a second student summed the indices and got 6^{5}; and the third went for 6^{4} because the indices increase in consecutive numbers. Note, none thinks that the base number is important in their considerations because 2 x 3 = 6, just as 2 x 5 = 10. This is a rich set of questions and comments that encompasses the meaning of indices, how to manipulate the equation, the laws of indices and numbers written in standard form. After one hour, Helen asked the year 10 class to reflect on how the inquiry process had contributed to a growth mindset. The advantages of inquiry include asking questions, using mistakes to learn, the development of fluency and thinking about what has been learned. Helen reported that the inquiry "really engaged some of my most challenging students and generated the best group work and discussion of the year."
Helen Hindle is head of the mathematics department at Park View School in Haringey (London, UK). She was an advanced skills teacher at Longhill High School (Brighton, UK) when she devised the base and index prompt and carried out the classroom inquiries. Helen runs the mixed attainment maths website. You can follow her on twitter @HelenHindle1. Resources Prompt sheet PowerPoint
 Online inquiry Robert Dale, a teacher of mathematics at School 21, used the base and index prompt to initiate an online inquiry during the 2020 lockdown. He reports on the development of students' reasoning.Over the span of 6 weeks, year 9 students engaged in a series of online lessons coupled with selfstudy. We began the course by trying to spot patterns in examples like 2^{2} x 5^{2} = 10^{2}, using mathematical vocabulary to explain what we could see. I then modelled to students, using a 3by3 grid (below), how to test further examples of this pattern. I encouraged students to "test the limits" by evaluating nontraditional examples, using decimals, fractions, percentages and surds. From this, students had a set of examples and nonexamples for the pattern. At this point, I introduced mathematical proof. For this, I modelled how to prove one of my own examples using algebraic representation, then, following this, students chose an example of their own to prove. After each lesson, students drafted and redrafted a short explanation about the patterns they had seen. They added to this throughout the course as they incorporated more examples and nonexamples, as well as their algebraic proof. Two lessons were dedicated to peer feedback: we did this using Google Jamboard. I asked students to focus their feedback around kindness, specificity and purpose. They then used this feedback to redraft their own work. At the end of the course, students were asked to improve on their final drafts in relation to an exemplar and assessment criteria before being submitted.
Read the students' mathematical journals here.
Robert Dale is a mathematics teacher at School 21, a 418 school in Stratford (east London, UK). You can follow Robert on twitter @RobertD87417849.
