I had other specific reasons for adopting an open approach to introducing this topic: one of the classes that I currently teach comprises of students who found their mock exams difficult; confidence and conversation is low. Recent lessons have been a struggle, with me enforcing rules to raise standards of homework and attendance. I wanted to bring some life back to the room, to get students enjoying mathematics again, to get students talking to me and to each other, to cede some authority.
They found the absence of clearly defined problems and lack of single correct answers difficult to deal with.
We talked about the difference between school mathematics, and what mathematicians actually do. We talked about how the problems a mathematician approaches are often ill-formed. We talked about how mathematicians have posed such questions over thousands of years, and how mathematicians work together in communities to solve these problems. We talked about how these communities of mathematicians decide what they think worth working on, for a variety of reasons (practical, aesthetic, economic, mathematical value, ...).
Although they found it difficult, the students came up with some interesting conjectures, which I wrote on the board without judgement:
Interestingly, some students revised their conjectures publicly as I wrote them on the board, such as C#2 on the right.
Crucially, one student - Toju - had independently researched the binomial distribution, which became 'Toju's method'. Here are her notes:
Toju and I shared this (excellent) research with the class. I then asked the students to explore the conjecture that attracted them using Toju's (binomial) method of expansion.
Of particular interest was C#5 on the right, (perhaps because it is the least believable / most exotic). This led to the realisation that the binomial expansion doesn't make sense with negative powers... students then went on to make connections with reciprocals and rationalising the denominator:
Some less confident students worked on checking C#4, which provided a means of simply practising the binomial expansion - subordinating practice to problems.
It was interested to me that very few students decided to explore C#1, because "they knew it was true". However, I still decided it would be valuable to discuss this conjecture with the class in order to highlight the binomial expansion process, especially with terms of the form (a-b)^n. In doing this, we convinced ourselves of correct statements for C#1 and C#2.
This is a critical part of this teaching approach: to know when to draw students attention to what is significant, to highlight the crucial content, within wider mathematical and social aims.
I sometimes used the board to make good ideas public. I listened carefully to what students were doing, but from a distance, resisting the urge to interfere where I was not needed.
Towards the end of the lesson, I decided to call a whole-class conversation, to bring ideas together, to address important issues, to highlight important ideas, to see what the group mind was thinking, and finally to set them a further task for homework:
I alluded to some questions I have been thinking about around multiplicative inverses and primes, by writing a few examples on the board: Are the primes in Z also primes in this number system? If not, what are the primes? What are the units in Z and in this number system - which numbers have a multiplicative inverse? Does every number have a unique (prime) factorisation in this number system?
As I left both lessons, there were students stood at the front of the board, pointing at what had been written, discussing what they were going to try next...