How do you introduce a new mathematical topic or concept? Do you give students a task to do, or do you start by explaining everything?
I think most teachers do a mixture of these things, depending on the topic and the class, and sometimes they orchestrate something that is kind of in between - what I call an interactive introduction. This is highly teacher-led, but aims to be more like a conversation and discussion than a monologue. This doesn't mean that it it is a 'free for all', in which anyone can just say anything that occurs to them. Nor is the teacher merely relying on one or two students happening to know what they wish to teach and telling everyone else. Realistically, only a few of the students will get to contribute orally to any particular interactive introduction. But, when an interactive introduction works well, all of the students will be equally able to 'participate' by engaging in the thinking process. They follow the thinking of the discussion, which is carefully planned not to depend on any knowledge which the teacher hasn't yet taught. And the teacher plans the interactive introduction to involve moments of puzzlement and surprise. The students are not left to figure out the content for themselves, but nor are they presented with it on a plate, all tidied up and complete. The teacher leads them to ask and answer the relevant questions.
It is easy to write a paragraph like that one, having my cake and eating it, and making it all sound so good. But how about some examples? Over time, many teachers have developed really nice ways to introduce topics, but I am not sure that these typically get shared so much. Teachers often share 'resources', which usually means either worksheets oriented towards the students - tasks for the students to do - or PowerPoint presentations for the teacher, that generally explain content and provide examples and exercises. Neither of these is quite what I'm talking about when I say an interactive introduction.
So, I'm going to share a few examples of how I have introduced certain common topics. I'm not making any claims for greatness here, and I'm sharing them as Word files so that you can cut and edit as you wish, if you find anything there that you want to use/develop/improve. I've kept each one to 1 side of paper, but hopefully there's enough here for you to see what I'm trying to do. Certainly, any kind of 'scripted' lesson has to be 'made your own' before you can authentically use it - I wouldn't envisage reading out any of this word for word, but instead attempting to capture the overall idea and adapting it to your own style and purposes. I've chosen the specific mathematical examples used in them quite carefully - certainly much more carefully than I could have done if you'd asked me on the spur of the moment to get up and explain something, unprepared. I think the particular examples might be the most valuable part of these interactive introductions, but please see what you think. I'd be very happy for lots of criticism of them in the Comments below. If you hate them, that's fine!
So here are:
1. A first lesson on 'standard form'. (I discussed this one in my most recent podcast with Craig Barton.) (Word and pdf formats)
2. A first lesson on 'enlargement'. (Word and pdf formats and the associated PowerPoint file.)
3. A first lesson on 'circles and $\pi$'. (Word and pdf formats and the associated PowerPoint file.)
And, finally, as a bit of a further experiment, I've also had a go at making a video of me introducing the idea of complex numbers (Word and pdf formats of the sheets). This is the sort of thing I would do with a sixth-form class in which I could assume that the students were familiar with the quadratic formula but have had no formal teaching about $i$. (You might also wish to see the related article, Foster, 2018.) Of course, in real life it wouldn't be a monologue like this, and would be 'interactive' to some degree. (And apologies for the sound quality on this recording - it turned out that the microphone wasn't plugged in, so it was recording through my laptop, but I didn't want to bother re-recording it!)
So, this is a shorter blogpost than usual, in order to give you time to look at the materials I've linked to.
Now over to you - comments, criticisms and improvements, please...
Questions to reflect on
1. What are your thoughts on the idea of 'interactive introductions'?
2. What comments do you have on any of these specific examples?
Note
1. You can listen to the episode here: http://www.mrbartonmaths.com/blog/research-in-action-16-writing-a-maths-curriculum-with-colin-foster/
Reference
Foster, C. (2018). Questions pupils ask: Is i irrational? Mathematics in School, 47(1), 31–33. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Is%20i%20irrational.pdf
I think these are excellent, engaging and fun. I work as a tutor and for many students I have taught, their main problem is not mastering a technique, but understanding that it relates to something they might actually care about! The standard form example seems strongest to me in this context. The complex introduction is slightly weaker in this context, in that it seems quite reasonable the x^2 + 3 has no solutions, in the same way we say 0x = 3 has no solutions - however hopefully with an A-level class the possibility it has solutions is exciting. Bombelli's "wild thought" at solving the cubic x^3=15x+4 in the 1500s (I saw it mentioned p5 in Tristan Needham's Visual Complex Analysis) where pretending root -1 exists suddenly allows you to find the real root 4, for me seems a stronger motivation. However this needs balancing against it being more complicated, though students can check (2 +- i)^3 gives (2 +- 11i) and confirm Bombelli's thought wasn't too wild. (Also, I haven't commented on any previous blog, but I have read them, and found them really helpful and thoughtful. Thanks for writing them.)
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