A Beginner's Guide to Heuristics

Hi, I'm Jane Gill and today I'm going to talk to you about heuristics. I believe they're the key to successful problem solving for children. I came across heuristics in the front of a textbook. It talked about relevant heuristics that children can use to solve problems and challenging questions. So I was obviously interested and thought as a math concern consultant, I really do need to know what heuristics are. I work a lot with teachers and they're always asking me, how can we help our children become more independent problem solvers? Rather than saying, I'm stuck. When they're faced with a problem, we want them to be confident mathematicians. That's what I'm passionate about. I want to start with a mathematical problem. And by asking you, first of all, what is a mathematical problem? I often ask this teachers, we're providing problems for children all the time. It's part of our curriculum. Solving problems, reasoning, fluency. What is a mathematical problem? Truly, though, and this is what I've come up with, a mathematical problem is something that you actually don't know how to do, not straight away. How often do we actually give that to children? A problem that they can't do, that they have to think about? And I think if you reflect on the problems that you give to children in your daily teaching, quite often we give them ones that they can actually already do. So are they problems? I think if we try and give children problems that they can't immediately solve, there are some benefits. It means that it will equip them for real life and maybe some test questions that come up in certain year groups. But there are barriers as well. Children will say that they're stuck, that they can't do it, they don't know what to do. So I want to start by thinking about this and helping you to reflect really on how often we give children something that they really have to think about. Let's consider this problem, not a very difficult one. And of course, problems can be difficult for some children and not for others. Again dependent on age and stage. But let's look at this one. Can we solve it straight away? Do we have to think about it? But what skills do children need? Jo buys three fruits. We can see the prices. Two of them are the same and he spends 23 pence. I wonder what he buys. When I ask teachers this and children, straight away, we think about the arithmetic skills that children might need. Yes, they might have to do some addition. I would imagine they might think about doubling, because if they remember Joe buys two of the same fruit and they might do some subtraction because they might work back from the total that he split. But actually, when I've done this with children in year two and year three, the first thing that you've got to do is decide how you're going to approach it, how are you going to start it? They could guess and check, they could make a list, they could act it out. I'll show you that later. They could work backwards. In other words, they need to choose a suitable heuristic or a problem solving rule. A problem solving approach that will help them solve the problem. I work with teachers a lot and we look at 11 heuristics. They're all beneficial. I think some of them are probably the ones that we use more often than others, but they're really useful for helping children solve problems. You'll notice that some of them are helping children to represent a problem, like making a list or drawing a diagram. You've got some that help children to make a calculated guess, like supposing something or, or guessing and checking. You've also got some that help children go through the problem, go through the process, like working backwards or maybe change the problem to help them see it in a different way. They're all useful heuristics and I've only got time to focus on a few of them. But I think you'll see that because mathematical problems can be solved by using a variety of approaches, it's worth helping children to see these different ones so that. So that they can make choices. The word heuristics comes from heereskine to mean to discover. We want children to discover their own way through a problem. If we're going to make our children truly successful at problem solving, we need to help them see a starting point in a problem. Often when a problem has multi steps and you can't quite see the answer yet, as long as you can find a starting point, you then can make a good approach. I found that if we frequently use heuristics with children, it will help empower them to be good problem solvers. It's recognised, it's accepted that you can teach heuristics. It's not something that just happens by chance. They're not routine steps. So we're not teaching a procedure like with a standard written method. We are helping children to make informed decisions. Let's consider this problem. Lara buys these items, it costs her 30 pence, a pencil and a ruler. Ali buys two pencils and a ruler and it costs him 42 pence. How much does a ruler cost is the problem. We've got to find out. Think about what heuristics children could apply to help them solve this problem. They could draw a model. Bar modelling is a very popular heuristic for children to choose as it helps them to see the wood for the trees. Or in this case, look at Lara's picture of her pencil and her ruler hiding in Ali's picture. That will help her see, help us see that 30 pence can be subtracted from 42 pence to find one of the pencils. Then we can label everything up and we'll be able to see what we've got to do to solve that problem. Or not all children will choose that one. Some children will choose to work backwards. In a sense, drawing a model helps you work backwards. But some children might not need to draw a picture in order to do so. They can just see straight away that if they start with 42 pence and take off 30 pence, it will leave them with the value of one pencil and then they can work through the equations until they find their missing number. For the price of a ruler, another heuristic we could use is guess, check, improve. I'm going to make a guess. I'm guessing that, and I think rulers are usually more than pencils. Let's say that the ruler would be 20 pence and a pencil would be 10 pence. This is a good guess. It's a guess a lot of children would go for nice easy numbers to add 10 and 20 and that makes 30 pence. That seems to work. Let's try it now into the other picture that I've got here. Two pencils and a ruler. Ah, it's not enough. Not quite enough. So what I'm going to do with my guess is I'm going to improve on it. So I've seen that it's not quite enough. So I'm thinking that probably the best thing to do would be to make the ruler a bit more. Now, some children might go a bit less because they haven't quite seen what to do to adjust, but they'll soon realize that that guess is getting worse. And so by checking and improving on your guess, it helps you see your way through the problem. So I've tried 11 pence look for my pencil, which would make the ruler 19 pence, because now I can use a bit of listing and I can see that that would still give me 30 pence, but it's still not enough. Look, when I get my 11 pence and 11 pence and 19 pence and add them together, it comes to 41. So I'm going to try again and this time I'm pretty sure that I'm going to be right and I'm going to guess 12 pence and check and see that it works. So guessing and checking is actually a really important heuristic to show children to use. Many children will say to you when they're not sure how to solve a problem, I'm just going to guess. But actually guessing is a really important heuristic as long as it comes with improving as well. It's undervalued because children think they're just using it because they don't know the right way in. But actually, if we make a guess, it always gives us insight into what else we can do, what else to try. And as you can see, it's often the work of working systematically. If I just go back to this one, look how by guessing and improving, we've now got systematic listing. So I can look for patterns and I can see that I'm getting there. Let's go back to the fruit problem that we had at the beginning. I'm thinking now, okay, I'm going to show children that trying something out is a good idea. So I'm going to make a guess. Joe buys three fruits. Two of them are the same. Let's guess, for example, that he buys two apples. Now I'm also going to use the heuristic, acting it out, because some children that I've come across on my travels still finding it difficult to remember all their doubles. We're working on it, but at the moment let's presume that I'm going to have to act out how to double six because I want to double the price of the apple. Here I've got my 10 frame on my baking tray that I use a lot in schools. And here I'm going to do my doubling six. So to double six, I'm going to just make a 10. So let me just make a 10. And I think this will help me see quite quickly that double 6 is the same as a 10 and a 2, which I know to be 12. Ah, that leaves me with an amount. Now, if children weren't sure what the missing number is, I can still use my 10 frame to see I've got 8. 3 more here look to make 23 in total. So that would give 11. But I haven't got an item for 11. So although I guessed that one, it didn't work out this time. So Let me try again. Let me go for seven. I'm going to guess seven because I guessed six last time. So I'm going to double seven. Now I'm going to act it out as well. I'm going to use that heuristic. Okay, I quickly act out what it is to double 7. It's the same as a 10 and a 4. 14. Aha. Let's have a think. Now what I've got left, I've got 6, 7, 8 and 9. Ah. And there is an item for 9 pence. So by using the heuristic guess and check and actually acting it out for those children that need that, I've discovered an answer that works. Now I'm intrigued because maths is all about pattern and it's very intriguing to think. I'm curious. Okay, I wonder if there are any other combinations that could work. So now I'm kind of looking at my listing. I don't really need to perhaps do the acting out anymore because I can start to see some patterns emerging that help me get my answers. I could. Even if there was something for 8 pence, I could put that in and have a look and see what this pattern would be. And noticing patterns is something that children can do very readily when they have listed systematically, when they have worked systematically. So actually, guessing and checking is a very valuable heuristic because it leads children into pattern spotting and making systematic lists. When I've worked with teachers on these heuristics, it's important to spend time practicing some of the key ones. So I'm going to share two problems with you now and ask you what would suitable heuristics be to actually solve these problems? There's probably more than one way, as we're already discovering, but I want us to practice a key one. So let's look at this. A tablet computer was reduced by a third in a sale to £90. What was the original price? And maybe a slightly trickier proportion question about Anne making 48 cookies, of which a third were lemon, three quarters of the remainder were chocolate and the rest were ginger. And that question is asking us how many ginger cookies she made. Or it could equally ask us how many of each type of biscuit. So let's just have a think about what we could do for that heuristic. Now, I'm a big believer in paper strips. I don't go anywhere without them. And here I'm going to just have a think about how I could also use acting it out. So ones involving fractions and proportional problems. You often can find that you can use paper strips really well and bar modelling lends itself brilliantly. So we're going to use the heuristic draw a model and act it out using a sort of practical bar model. Now, because this problem, let's start with the tablet computer one, it says it was reduced by a third. So I first of all want children to get at their practical bar model, unfold it into thirds. Well, that can be quite hilarious actually. I've seen children trying to fold paper strips into thirds. So it's a really good skill to practice. A group of year six children recently thought, well, they'll fold it in half and half again. And I thought, I'm sure they'll know that's quarters. They did and then they lopped one off the end, which was quite interesting. Very good problem solving way of getting thirds. But if we get our thirds because we're trying to get equal parts, so. So we're folding it into thirds as best we can. So we have our thirds, we can then open it up and actually look at this problem in a practical way. A tablet computer was reduced by a third, so it's been reduced by a third. So I'm going to fold one of the thirds away and it now costs, look, £90. That's when I can see that the £90 that it costs are two of the pieces, not the three like some children might see if they don't draw a model. And therefore I can now work out that half of 90 is 45. So I need to label. If I was drawing it as a bar model like this, though, I could label each piece 45 to see that my original price must be three of these 45s 135 pounds. And it would help children very much get to that equation. When we're working on bar models as one of the key heuristics, I'll always ask children, what do you know? What don't you know? What's the part? What's the whole? Draw what we know, Draw a model, label what you know already and then it will help you see what you're yet to know. Let's look at this one. Anne made 48 cookies. I'll keep with this because it's still about thirds to start with. Not and made 48 cookies. Here they are. A third were lemons. So I'm going to fold that third away and remember that they're lemon. Okay. Three quarters of the remainder were chocolate. I've got two pieces. Well, that's easy. I can just fold those into Quarters and the rest were ginger. How many ginger cookies did she make? So here's my third folded away for the lemon ones. And I can now see the equal pieces that I've got left. Obviously with a bar model, children can draw this as well. To see it, a third of 48 you might find is 16. Or you might now see these six equal parts and realize that each of these parts is eight. And so she must have made eight ginger cookies, 24 chocolate ones, and her 16 lemon ones. Acting it out, drawing a model. A really good heuristic to practice. And that's a really difficult problem that you'd find in secondary school. And yet children in year four can solve this problem because they have a model to use. I think of all the heuristics and I shared 11 in the poster that I showed you. Drawing a model, guessing and checking, obviously with improving and making a systematic list are the three that I would definitely devote time to. I would do a series of skill based lessons that have mixed problems because let's face it, children don't often come across mixed problems and yet that's what they're faced with in real life. But the lesson would be about the skill of guessing and checking and improving on your guess or drawing a model and helping you see and represent the problem, making a systematic list to look for patterns. So teaching heuristics for me helps children to be confident problem solvers. It helps develop an open mind. There'll be children that use one method and children that use another, but they can still get to the answer. They can have that eureka moment, which of course, if you think the word eureka that we use when a little light bulb comes on and a child's discovered the answer to a problem for themselves is actually also from the same word. Heuriskind Heuristics. It's from those two words. It's the same root. And so I really feel that this is something that children can take forward and become confident problem solvers, independent problem solvers, and to just really enjoy mathematics.