How open should an open investigation be in maths?

I’m sure loads of people can remember as a child asking their mum or dad to keep the door slightly open at night. I can just imagine a maths teacher as a parent saying in response, “Exactly how far do you want me to leave it open, would 15 degrees be appropriate?”

How does this relate to open investigations in maths lessons? I’m sure that we all agree that open investigations can be very beneficial as a way for students to discover mathematics for themselves. Another added benefit is that if we can pick a question that allows all of our students to access the question whilst at the same time enabling the high ability students to take the problem further, then we’re onto a winner! We’ve all seen the disadvantages however when students lose focus when the access level is too high or they aren’t given enough guidance.

I recently set a  `reasonably` open investigation involving circle theorems to my top set year 9 class and since then have been reflecting on the “openness” of questions in mathematics. I rarely set very open questions such as the ones below:

1) Powers of 3 are 31, 32, 33, … Investigate.

2) Investigate right-angled triangles with whole number lengths.

3) A shape has a volume of 216. Investigate.

All of these three investigations fall at different levels on the “openness scale” but in my opinion they’re pretty open. I’m wondering how often teachers use open investigations such as these in their teaching? How much guidance would you give when introducing these investigations? To what extent does it depend on the age and ability of the learners in the classroom on how open your investigations are? Should they all be reserved as extension for high ability learners? Indeed, are they even appropriate for this purpose? Does anyone have any different open investigations which they think are `better` than these? Using the door analogy, is there a critical angle at which too much light streams in and the child can’t get to sleep?  The questions just keep streaming in. Anyhow, I’d be really interested in peoples’ opinions on using open investigations as I’m thinking that it could be a possible MA Dissertation topic.

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