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More tips from Craig Barton
Video transcript
hello my name is craig barton and welcome to this tips for teachers video now for much of my teaching career my focus has been on teacher teacher-generated examples some of you may be familiar with my work on myvariationtheory.com website where i put a lot of time into trying to create sequences of related examples so students can attend to the critical differences between them and notice key things about mathematical structure and so on however and i’m quite ashamed to admit this only relatively recently have i become aware and a massive advocate of the power of learner generated examples now for me in mathematics anyway the seminal work on this is the wonderful thinkers publication by the atm now within thinkers there are lots of structures for helping students come up with learner-generated examples in this video i’m going to focus on the two that i think are the most powerful because of their versatility i think you can pretty much use these for any idea in mathematics and i wonder if you’re a teacher of another subject whether these same structures can be applied to help students generate examples within your area okay now the way we’re going to run this is you’re going to be my students and i’m going to be the teacher so you can generate your examples as we go through this and as we go i’ll touch upon a few key elements of pedagogy but at the end once we’ve gone through the two structures i’ll just point out four key features that i think are important to help this run as smoothly as possible okay so here’s the first structure it’s called give me an example of and another and another and it works like this right on a piece of paper what i would like you to do is give me an example of a fraction that’s equivalent to two-thirds write that down for me please now you might want to pause the video if i go too fast or you’re getting annoyed or something like that right have you got that one okay on that same piece of paper what i would like to do please is write me another fraction that is equivalent to two-thirds okay and finally on that same piece of paper you know what’s coming right give me another now once you’ve got your three fractions and as i say feel free to pause if i’m going too fast here i’d like to do one more thing please and that is just spend a few moments thinking about how you constructed those three examples and here are three prompts to help you so how did you come up with your first example what was your technique there did that technique change when you came up with your second and third examples and did this process get easier or harder as you came up with your three examples just spend a few moments just contemplating that and pause the video okay now let’s jump into the classroom how i would run this so i do exactly the same as i’ve done with you there the only difference being as i’ll talk about a little later is students will be writing this on mini whiteboards at the end of this process once the students had independently come up with three fractions and then spent a few moments considering how they constructed their examples i would ask them to hold their mini whiteboards up all at the same time so i could see the spread of ideas that they’ve come up with then what i’d ask students to do is to discuss their examples with the person next to them first show each other their three examples and see if they’ve got any the same or any different but then what i’m particularly interested in is them describing their process for constructing their examples and by prompting them to do this independently first before the discussion i really think it helps make that discussion much more positive and much more productive and then of course what i can then do once students have had this discussion is having seen the range of examples that are out there i can then instigate a whole class discussion so i can say miriam i see your first one you came up with was four sixths who um how did you come up with four six for that first one michael but you didn’t do four six you did you did 40 30 40 60th or whatever it may be and then i can just use what i’ve seen when i’ve seen all the whiteboards to pick out particularly interesting or quirky examples and we can discuss them of course the thing i’ve forgotten to mention is that when students are having this initial paired discussion they can of course check each other’s examples so they get additional practice of fraction equivalence or simplification and so on and so forth now what you may have noticed is as you’re challenged to come up with another and another perhaps you start doing a slightly different way of thinking perhaps everyone has this first obvious fraction that comes to mind when they do two-thirds maybe it is four-sixths but unless you’re asked to come up with another and another then maybe you start to think slightly differently some students may just keep doubling double the numerator and denominator then double it again but within a class of 25 30 students you’re bound to find some different approaches and that’s going to really add fuel to the whole class discussion that you can have okay so that’s the first structure the second one is called additional constraints now i think this is a little bit more challenging both in terms of challenging to write as a teacher but also more challenging for the students so should we give this one a go all right here we go so we’re still on fractions about a slightly different skill this time so i would like you to think of a pair of fractions that add to one and again can you write that down any two fractions that add to one and again you might want to pause the video at this point uh just so you have a bit of time all right once you’ve got that now i want you to think of a pair of fractions that add to one but also have different denominators and once you’ve got that as i say keep pausing me here now i want a pair of fractions that add to one have different denominators and have the same numerator and then when you’ve got that one i want them to add to one have different denominators and the same numerator and one fraction has got to be improper and once you’ve come up with those i want you to think about how you constructed your examples okay so again feel free to pause the video until you’ve had an opportunity to think through those yourself but now i just want to jump into the pedagogy of how i would run this this would be one where again on mini whiteboards but this time i would ask the students to show me their examples at each stage of the process in the previous structure it was the students wrote all three down before they showed here i want to see as we go through that’s really important because as this gets more complex i just want to pick up just in case and there’s any misconceptions happening in the class but also and i want students attention focusing on each of the challenges i’ve asked them to do at that moment in time so let’s say for example there was a really interesting answer that i saw for this second prompt well if i’m only seeing that after students have thought about all four prompts it’s quite hard to bring their attention back to that second prompt so what i like to do is give the students time to do this on their own so think fractions add up to one time to think about how they constructed their example and then i want to see everybody’s and then i can then go into the crowd and say oh how did you come up with that how did you come up with that and so on and then we move on to this second uh constraint so this um as i say is a little bit more difficult to write because you’ve got to kind of i mean i spent ages doing this one and it was a i came up with about three or four impossible ones which again would have been an interesting twist but perhaps perhaps not ideal uh but i and maybe these are more difficult for students to answer but i think these this is a super super super useful structure so um i mentioned just a few kind of pedagogical points that i wanted to make and the first is i think mini whiteboards are essential for this they’re essential for a number of reasons and it’s really important that i as a teacher can see students answers and for me a mini whiteboard is a really really powerful way for doing that means i can pick up on any misunderstandings but also i can then clearly see really interesting answers that i want to use in the in the whole class discussion it’s also really good for the students as well we know students like to kind of rub off things and they’re they’re less like less likely i think to fear making mistakes and if it’s kind of rub offable if that’s even a word and but i also think it works well in the paired discussion it’s much easier i think to swap a mini whiteboard with your partner than it is to go through the hassle of swapping books and then you know and so on and so forth i just think it’s a lot easier and kind of practically for students to work together to have peer discussions with minnie whiteboards and that leads into the second thing and really greatness for peer assessment and peer discussion peer assessment in terms of checking each other’s answers that’s a really important part of this but also peer discussion and because students have been working on the same task but may well and i think it’s quite likely this have come up with different answers they can have some really good discussions and about their answers particularly if they’ve been prompted to consider already how they came up with their examples final two and again i won’t use these all the time but they you may find these useful and you could say to students maybe in that give me an example and another and another it could be give me an example then the second one could be give me an easy example then the third one could be give me a hard example and it just adds an extra little twist maybe an extra little constraint in there that perhaps gets the students particularly if you’re finding they’re just kind of doubling numerator and double in the denominator each time maybe that prompts them just to think a little bit differently it’s one of those that’s worth experimenting with likewise i really like this give me interesting examples so again you could use that in the give me an example of and another and another the last one could be give me an interesting example and i find it really fascinating to um to see what students find interesting and that can be a really worthwhile discussion or you can use it throughout those additional constraints you could say to students okay if you found an example really quickly i want to come up with a second example but make it as interesting as possible so in the additional constraints one it’s not just giving me two fractions that add to one it’s giving me two fractions that add to one and then give me an interesting pair of fractions that add to one and so on and so forth so anyway they they are two of my favorite structures for learner generated examples what do you think of those have you used those before do you use those slightly differently uh to id i do if you teach a subject other than maths could you make use of those um in in your subject i’d be fascinated to to to hear how you do that if you found this in any way useful i’d be so grateful if you could like this video and subscribe to the tips for teachers youtube channel and if you want to visit tipsforteachers.com uk you’ll find loads more tips like this not just from me thanks so much for watching