Base and index inquiry
Mathematical inquiry processes: Make connections; test different cases; infer and explain rules. Conceptual field of inquiry: Base and index numbers; square numbers.
Helen Hindle devised the prompt to generate a short inquiry about the relationship between base and index numbers. The product of two and five is 10, but will that work if the numbers are squared? Or cubed? The prompt could form the basis of a structured inquiry. Students verify the truth or otherwise of similar equations and generalise.
Helen Hindle trialled the prompt 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 22 or 52 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 22 = 4, 52 = 10 and 102 = 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.
52 = 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 right-hand corner of the board we can see three students speculating about the law of indices: one claims 22 x 33 = 66 because you multiply the indices; a second student summed the indices and got 65; and the third went for 64 because the indices increase in consecutive numbers.
No-one has suggested 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."
Robert Dale, a teacher of mathematics at School 21 in Stratford (east London, UK), 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 self-study. We began the course by trying to spot patterns in examples like 22 x 52 = 102, using mathematical vocabulary to explain what we could see. I then modelled to students, using a 3-by-3 grid (below), how to test further examples of this pattern. I encouraged students to "test the limits" by evaluating non-traditional examples, using decimals, fractions, percentages and surds.
From this, students had a set of examples and non-examples 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 non-examples, 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.