Struggling students are exploring too
I firmly believe that all students deserve to play with mathematical ideas, and that extension is not just for the fast or "gifted" students. I also believe that you don't necessarily need specially designed extension activities to do exploration – a simple "what if" question can easily launch a standard textbook exercise into an exploration.
This is lovely, but one problem is those students who on the face of it don't want to play. The majority of students I work with in the MLC are not studying maths for maths's sake, but because it is a required part of their wider degree. That is, I am helping students who are studying maths for engineering, or calculations for nursing or statistics for psychology. A lot of these students just want to be told what to do and get the maths over and done and don't like "wasting" time playing with the ideas.
Or so I thought. I have realised recently that actually they do like playing with the ideas. I just couldn't see that this was what they were asking for.
One of the questions I like to use to play with maths is "what if?" To ask it requires the asker to notice a feature they think might be different, and wonder what would happen if it was, and so investigate how this feature interacts with the other features of the situation. This investigation of how ideas interact is exactly what mathematics is, to me. I am very very used to doing it at extracurricular activities like One Hundred Factorial, and it's easy to do with students who are doing very well with their maths and have the breathing space to wonder about this stuff because they've finished their work. For students working on an assignment they are really struggling with that they wonder about the usefulness of for their degree and which is due in the next 24 hours, it's not so easy.
Only maybe it's a little easier than I thought, because I started to notice that the students were already asking questions about the connections between things.
A very common question students ask around exam time is "What would you do if the question was like this ...?" as they suggest a change in a small detail that makes it more similar to the past exam question they have at hand. I used to get really annoyed at this sort of question, but then I realised that in order to ask this question they had already noticed that the two problems were similar in most respects and different in this one. Sure, they may be motivated by a belief that success in maths is about a big list of slightly different problems and remembering ways to deal with each, but on the other hand they have noticed a relationship and are trying to exploit it, This is a mathematical kind of thought, to look for similarities and differences and relationships, and we can hang on tight to it and actually learn something!
Another kind of question students ask is "Why is this here in this course?". I used to get annoyed at the whinging tone of voice here, but then I realised that a student is begging for a connection between this and the rest of what they are studying, and connections is precisely what understanding maths is all about. I can respond to it by saying that yes I am also confused about the curriculum writer's logic for including it, but then we can search out connections together to see if we can find them.
A third kind of question is the one where the student looks at the lecture notes or solutions and asks "How did they know to do this?". Sure, it's usually motivated by wanting to be able to successfully do it in their own exam in restricted time, but on the other hand it does recognise that there ought to be reasoning involved in that decision, as opposed to just guessing. This expectation of reasoning is the beginning of believing that they themselves could reason it out too.
My second-last kind of question is "Why is this wrong?" as the student points to the big red cross on their MapleTA problem on the screen. Sure they just want to make it all better so they can submit the damn thing. But they are also recognising that there must be a reason why it's wrong, and so are looking for meaning. These students are often ripe for the experience of looking at the information and how it's related to what they've entered, thus doing exploration. They are also usually ready to try various different ways of entering the result, or different strategies of getting a right answer to see where the edges of the idea are.
The final kind of question I want to mention isn't even a question, it's an exclamation of "It doesn't make sense!" Even this is telling me that the student thinks it ought to make sense. They are crying out to do something to make sense of it. Often they describe being almost there and needing something to push them over into understanding. This student is ready to explore the edges of the understanding they do have to see where the nonsensical stuff can fit, or where their ideas need some tweaking to fit together better.
It is very helpful to me as a one-on-one or small group teacher, or even a lecturer with a big room of students in question time, to be able to see the questions struggling students are asking as cries for sense-making exploration. It doesn't matter that they are struggling or don't understand things. Indeed, being in a situation of not knowing is exactly what we are doing for the fast students when we give them extension activities, so why is the everyday maths the slower students are not understanding any different? In my experience students like being treated as if they are behaving like mathematicians when they have these struggling sorts of questions. And it was so much easier to treat them that way after I realised they are behaving like mathematicians when they have these kinds of questions.