More tips from Craig Barton
Video transcript
hello i’m craig barton and welcome to this tips for teachers video now i’m going to be honest with you the next couple of minutes are going to be slightly painful for me because i’m going to have to admit yet another of my many mistakes as a teacher so let’s imagine that you’re my lovely students and my job is to try and explain to you the concept of a triangle now if you were unlucky enough to be taught by me early on in my career then no doubt i would have started with a definition because i thought that was the best way to understand something but the problem of course with definitions is that often you have you end up having to define terms within the definition to understand the definition itself so to understand what on earth this is banging on about you’ve got to know what on earth a plane figure is what straight means and what sides mean whatever you’re defining a definition it’s not looking good for students understanding it so fairly quickly i realized that wasn’t the best starting point but where i moved to wasn’t that much better because what i then started with was a single example and worse than that it tended to be a fairly conventional example now the problem with conventional examples is that students it ends up narrowing students understanding an experience of a concept and they start making erroneous assumptions so if you take this example of a triangle if my example of a triangle is this and my subsequent examples when i start doing angles in a triangle and all that all look like this as well then students may start to assume that all triangles have horizontal bases all triangles are isosceles or equilateral and it becomes really problematic so after a while i realized that wasn’t so great so then i started trying to widen my students appreciation and experience of in this case the concept of a triangle with multiple examples so this is a bit better because now we’ve got an orientation change we’ve got a different type of triangle a right angle triangle and so on but of course this is still severely lacking because what hit me and it took about 10 years for this to hit me was it to truly understand something it’s not enough just to understand what it is you’ve got to also understand what it isn’t and hence of course the power of non-examples so after about 10 years of messing this up i started to introduce examples of something that is and something that isn’t and this was certainly better but i still think we can improve upon this so when i’m trying to explain a concept to students now i have three principles that i try to adhere to so the first is as we’ve just discussed there i try and sequence a mixture of examples and non-examples but i also do two other things so the first is i include boundary examples examples right at the concert right at the boundary of where that concept is or that concept isn’t i’ll show you an example of that in a second but the third thing i do is perhaps the most interesting and that is i try and make consecutive examples related so students can attend to the critical feature now let me try and demonstrate this with a sequence of examples on triangles and then i’ll also do it for another mathematical sequence but what i’m interested of course if you’re not a maths teacher is could you do something similar for your subject and what will you need to change to make this work so let’s have a look at triangles let’s imagine you’re my students this is what it would go like you’d ideally have mini whiteboards in front of you and i’d start with a conventional example like this something that you’re pretty familiar with i tend to start with an example versus a non-example just to give kind of that early sense of success so i put something like that up on the board and i would state very proudly and i’d write it down as well this is a triangle and then what i do is i show you a second example and i’ll prompt you to pause and silently look at that and consider what’s changed and what stayed the same now early on when i’m teaching this process to my students i may then kind of go into the crowd ben what have you noticed then ben may say ah the orientation’s changed at this triangle it’s flipped over something like that but hopefully soon students will automate this routine so i won’t need to prompt them to do that and then what i’d state once students have attended to the critical feature that’s changed i would state that this is still a triangle and then i pause again to get students to reflect on that what have they learnt there about what it takes to be a triangle well hopefully they’ve learned that the orientation doesn’t matter was that around wherever you want you’ve still got yourself a triangle all right so now what i’m going to do is this so i show my students that image and i pause and that will be their cue to reflect what’s changed and what stayed the same and then i would then reveal this is not a triangle and then pause again and that pause that second pause is so important because that’s the prompt the cue for students to try and explain what’s happened what critical features changed and what impact has that had on the concept so the critical feature between one and two didn’t change it was still a triangle when the critical feature between two and three changed it did change whether it fit inside the family of being a triangle and then i’d carry on so okay what critical features change now well now we’ve no longer got straight sides we’ve got curved sides what’s happened there still not a triangle so each time i’m showing an image pausing so students can reflect on what’s changed and what’s not changed then revealing the answer of whether it fits into the family of the concept or not and then pausing again to get give students an opportunity to see if they can reconcile what’s happened i’ll show you a few more so change the orientation that’s a triangle now what am i going to do well i’m going to make sure i’ve got both concave and convex curved lines just to try and cover the the full scope of them of the domain of this no that’s not a triangle then i’ll do something like this so again just to recap i’ll show students that i get them to pause and think what’s changed well now the lines are straight but wait a minute we now seem to have a little kind of bend in this is that a triangle or not once once i’ve reflected i’ll reveal no this is not a triangle then pause again now at any stage i’m free to go into the crowd if i sense students are confused a lack of understanding i can just stop things and say okay let’s focus on question seven mirren what did you notice about question seven oh fantastic it’s got four sides okay what does that tell you about a triangle everybody write it down in your mini white balls so i’m free to do that at any stage or i can just let this sequence run and i’ll end with a triangle that’s a little bit weird and non-conventional triangle and that is a triangle now how do i wrap up this process this only takes a couple of minutes well at the end of it my two favorite things to do to students are to say okay the first thing i want you to do on your mini whiteboards is draw an example of a triangle that someone might think was not a triangle and then secondly draw a non-example of a triangle that someone might think is a triangle so what i’m trying to do there is really assess students understanding of this concept not by saying draw me something that’s obviously a triangle and obviously not because you just draw you know an equilateral triangle in a circle there i want them to go right to the boundary and then i can have a look at their mini whiteboards when they hold them up they can swap with each other and i can get a real sense of their understanding of a concept so that’s one way of doing it another example might be something like this i wanted to do one that’s non-visual so let’s take an expression so again imagine you’re my students and i’m trying to teach you what it takes to be an expression so i’ll start with this i’ll show you five and i’ll say that’s not an expression then i say right okay what have i done there just pause what’s changed and what stayed the same so in students heads it’s gone from a number to a letter i’ll then reveal still not an expression okay all right what am i going to do next i’m going to put those two things together so notice i’m not choosing examples from random here they’re always related that’s important so students can observe what’s changed and then observe the impact okay five plus x pause that is an expression okay pause again so students have an opportunity to start building this picture of what it takes to be an expression then i do this so students may be thinking after question three okay right so if you’ve got a number in a letter and an operation you’ve got an expression but when they see this no that’s not an expression now if they’re confused at that point they’ll probably say to them look okay stick with it let me show you a few more then we’ll have a discussion about it because i know what’s coming i want to build this up for students okay now let’s do a division is that an expression multiplication wasn’t maybe division isn’t either yeah okay that’s not an expression so what have i got to do next of course i’ve got to do subtraction that is an expression so at the end of that run there students may have a fairly decent idea of what it takes to be an expression but here’s the key thing there are still gaps in their knowledge at that point gaps that i can help them plug by carefully varying what comes next so again it’s a worthwhile um exercise here particularly if you’re a math teacher just pausing here and thinking what if students learn here but what’s still lacking what are some of the expressions that they could be presented with that they may miscategorize so let me show you where i went with this firstly i realized that i didn’t have any negative variables here so i just wanted to make a negative variable so just flip 6 around still an expression but then of course i’ve used x all the time i think math teachers are often guilty of that i don’t want kids thinking expressions always involve x so let’s chuck a y in there have i messed things up now nah still in the family of expressions things are looking good okay what happens next everything’s been linear so far let’s square it still an expression yeah still an expression okay so ever with each one of these students understanding is deepening and deepening as their experience widens what happens next well i noticed here i always had a single variable let’s chug a p in does that mess things up because without that example students may leave thinking expressions only ever have one letter in there that still got an expression there now i thought this was important as well every single one of these that’s been an expression has a number involved so i don’t want students making that erroneous assumption so let’s chuck that in as well and then finally of course let’s chuck an equals at the end we’ve broken it no longer and no longer an expression so going through that process again only takes a few minutes it’s pause when students see an example to see what’s changing what stayed the same and then pause after the reveal of whether it fits into the family or not so they can try and reconcile what’s happened and then the final challenge at the end an example of an expression that someone might think is not and not an example of an expression that someone might think it’s to push their understanding to the limits so just to recap there they’re my three principles i use when i’m trying to explain a concept to students a mixture of examples of non-examples boundary examples but then also making the examples related to each other so students attention can focus on the critical feature now that’s just a couple of examples from maths if you’re a non-maths teacher watching is that useful at all do you do something similar for that could you make that work what would you need to change to make it work let me know um if you found that in any way useful i’d be so grateful if you could like the video and subscribe to the tips for teachers youtube channel it really does make a difference and finally visit tips for teachers.cod.uk for more tips like this thanks so much for watching
2 replies on “Use related examples and non-examples to explain a concept”
Hi Craig,
An excellent video, really helpful to us as I am planning and delivering whole-school (EYFS – Y6) PD on explanations and modelling. The examples (!) of examples and non-examples is particularly helpful and 100% is useful and applicable to non-maths teachers as this is not regularly part of teacher’s pedagogy.
Thanks again!
Thanks Mark! So pleased it is useful.