Transcript of teaching mathematical reasoning
Speaker 1: Well good afternoon ladies and gentlemen and welcome to this very special edition of SyllabusPLUS. Special for two reasons. Firstly, the content today deals with an aspect of the syllabus which is arguably the most challenging. That is, teaching students mathematical reasoning, and special for a second reason in that we have a guest presenter who is well known to all of you. A man without peer in the world of mathematics education. Of course, we're talking about Peter Gould. Welcome Peter and thank you for joining us. Peter has been [inaudible 00:00:41] away temporarily from his important work in the early action for success initiative, one of the key programmes in the DC at the moment. We are very fortunate to have Peter this afternoon and also very grateful.
Peter, are you there? Wonderful.
Peter Gould: I'm just mastering clicking the right spots.
Speaker 1: Very well.
Peter Gould: I think I'm away.
Speaker 1: Wonderful. Okay, well I'll just change the presentation view, switch off my microphone and you're away.
Peter Gould: Okay well, the key thing with this is I wanted to talk a little bit within the time about what I think is the essence of trying to get the new syllabus into place, and something that at the same time is not that easy. That's the challenge of teaching mathematical reasoning. He says. Finding the right button. Much better. I wanted to start by talking about one of the issues and that's actually to do with beliefs, and the beliefs around what does it mean to do mathematics, to learn mathematics and then the associated thing of what does it mean to teach mathematics.
One of the interesting things with that is of course that many students believe learning mathematics is about the quick production of answers. 'Quick, can you tell me this?' They believe that the role of the teacher is to pass along procedures and the student's job is to apply the necessary rules. Many also believe there's only one way to solve a problem. The background sometimes is what do students believe around what doing mathematics is?
If that's what the students believe, many students believe learning mathematics is, what are the other views of mathematics? For generations, high school students have been studying something in school. It's been called mathematics, but often it's got very little to do with the way mathematics is created or applied outside of school. In fact, if you go back to the origins of the word mathematics, people often have that view so strongly that mathematics is about procedures, they get a bit surprised to find out that it comes from the Greek, and the Greek meaning of the word actually means 'learning' or 'study'. It's quite general around what mathematics is. It just means 'learning'.
So what's mathematical thinking? One of the things that's a bit surprising is the widespread utility and effectiveness of mathematics come not just from mastering specific skills, topics and techniques, but more importantly from developing the ways of thinking, the habits of mind used to create the results. That's actually a quote from an article that we've added to the website which is about habits of mind as a different way of thinking about organising curriculum in terms of high school's mathematics. Rather than just simply talking about learning the particular skills and topics and techniques, more about how do you actually think about problems.
From the syllabus, if you actually read the syllabus itself, it starts off by saying 'Mathematics is a reasoning and creative activity'. I bet you if you asked your students what mathematics is, that not many of them would come up with that as the first part of describing it, but that's actually what's in the syllabus. A reasoning and creative activity. It also talks about reasoning in the syllabus as a capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Again, that's a statement around it but sometimes I think when you end up giving a whole list of things, you're often struggling to define what you mean by something, because proving is a key part of reasoning in mathematics, but it's not the only component that people would think about within reasoning, and we'll come to that a little bit later.
Students are reasoning mathematically when they explain their thinking, deduce and justify strategies used and conclusions reached and explain their choices. Again, these are straight out of the syllabus. But when you think about communicating reasoning, students explaining their thinking is not necessarily nor uniquely reasoning in mathematics. I've often had students explain their thinking and I'm pretty certain it wasn't reasoning mathematically. Communicating mathematical reasoning might actually need a term teaching focus, and my reasoning with that is that writing up your reasoning is much easier if you can actually carry out the necessary reasoning in the first place. If you have trouble doing the first bit, writing it up is the least of your difficulties.
There are two difficult questions associated with reasoning. One of them is 'How do you teach students to reason mathematically?' The second one is 'How do you teach students to communicate their mathematical reasoning?' I'm going to address those, trying to focus particularly on examples around the first, and seeing how we go with touching on the second, because as I said previously, if you can get them to reason first, being able to get them to communicate their reasoning is less challenging.
Within the syllabus, you've got that progression. You've got the progression that deals with stage two outcomes in terms of reasoning, 'Check the accuracy of a statement and explains the reasoning used', all the way through stage 3, 'Gives a valid reason for supporting one possible solution over another', 'Recognises and explains mathematical relationships using reasoning', in stage four, and it is there that I want to focus today. In 5.1.3, 'Provides reasoning to support conclusions that are appropriate to the context'. Similarly in 5.2.3, 'Constructs arguments to prove and justify results', and then of course 'Uses deductive reasoning in presenting arguments and formal proofs'.
Let me go back to my focus for today, and I want to emphasise the stage four component which is around 'Recognises and explains mathematical relationships using reasoning'. In fact, if I had to say which one of those do I really want students to be able to do, if I could only get one I'd be quite happy with the stage four one. 'Recognises and explains mathematical relationships using reasoning'. Now, how might we teach our students to do this? As I said, I'll try to keep my examples largely coming from stage four, because it's not all that hard to see how they then develop into stage five.
This is where you get to do a little thinking. Not too much, but just a little bit, it is late in the afternoon. What we've got is the equivalent of a calendar month in terms of numbers, and the question that I'd like you to answer, I don't ask an awful lot of rhetorical questions, is which has the larger total, the three numbers covered by the yellow strip, or the three numbers covered by the red strip? This is not silence this is you working out what you think your answer is. Which has the larger total? If you've got a group of people around with you, you can start a [inaudible 00:09:35] going. Remember I'm still talking about three numbers covered by the yellow, the vertical, or the three numbers covered by the red, the horizontal.
All right. For those risk takers who've had a go at it, my next question and the question that's perhaps more critical is why? I can see that we've got an explanation in the specific type here. What I'm doing is modelling the sort of questions that I would ask with a Year 7 class to get them started with this idea around reasoning, and in this case ... I want to also mention now the idea about how do you deal with communicating mathematical reasoning, and the first one I'd say is you always start with convince yourself. If you can convince yourself of whatever your answer is, that's a good place to begin, but that's not where you end if you're going to deal with communication. Communicating with yourself is important, the next step is to convince a friend. Now I use the term friend because I use that with students, and it's the idea that a friend is more likely to believe you if you come up with a half way reasonable argument with things.
Now once we've got the context of the problem, and I've cut down the animations a little bit in this, because sometimes I can actually put up the strips individually like band-aids and cover them, I've shifted it a little bit now so I've got a red component and a yellow component sitting behind it, and I've got three numbers going across to the right. What would the total be? So I get some explanations and discussion going on at the moment which is good, because if I see that in a class it's really useful. However, the key component is being able to convince others, so it's good to actually hear a little bit of 'How do you know?' 'Why do you say that?' 'Does it always work?' going as well.
So the red total. What would the red total be? There, nifty discussion. Yeah, I think 54 sounds like a goer to me. And you can see how you get asked these questions within a class. If you need a more experience, shift the cross, shift it somewhere else and try it and work it out, and the discussion that I think I picked up a little bit of before is 'Is there another way to calculate this?' Because sometimes people will start with the idea of literally working out what the three digits are, or the three numbers are under there and adding them together. But there is a faster way to do it, and I know that you know how to do that so I'll leave that and move on.
Now, the question that's really quite interesting from my point of view is 'Why does it work?' Why does it work that you'll always get the same total under the yellow and the red regardless of where on that particular setup I move that strip? And I won't expect an answer now because I'm just monitoring and managing time a little bit with this as well, but the discussion is around why does this work, because we've got finding a specific example, doing a bit of arithmetic to convince yourself, but then 'Is there another way to work it out?' And then 'Can I use that idea to work out why it will always work?'
In stage four, we've got the idea or the outcome 'Recognises and explains mathematical relationships using reasoning,' one that I've mentioned before, 'Communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols.' Now, the moment you start to explain your reasoning with that particular problem, you're going to end up using communication and hopefully appropriate terminology, diagrams and symbols, because it doesn't take too long before you start doing them and explaining them. Generalizable numerical expressions play an important role in developing algebraic reasoning. Now when I use the term 'algebraic reasoning', for me algebraic reasoning is thinking logically about unknown quantities and the relationships between them. In high school maths, it often involves xs and ys but those of you that have done linear algebra in other places realise that it's about the structures generally, so it's often reasoning about unknown quantities in a broad sense.
Let's see if we can use some of that algebraic reasoning. If I have this particular sum, is there a way that you can work out quickly what that total is? And of course the 'quickly' is the thing with that. I don't actually expect you to add all of those numbers together. All right, and I've got an explanation that 'Yes, there's a quick way, I can do seven times 45', and there are different ways but the question to follow it up with is 'Why does it work?' And if I didn't want that but I think about it going down this way, which is very similar to generalising the previous problem, I think you get the idea that yep, I can do these totals in different ways.
Now I actually often think about this in terms of symmetry. I know that when we use the term symmetry most of the time we're talking about geometric arrangements, but mathematically it doesn't mean that, it just means that there is a sameness, there is a symmetry to the problem, and these are quite symmetrical distributions. If you think about this, you all thought about it from the point of view of somebody engaged in doing the problems. If you were working with this with your students, what sort of reasons would you accept as to 'Why does it work?' You don't need to necessarily write those up because it might take a while, but I want you to simply think about engaging with that. What's acceptable reasoning when children do it in your classroom?
Now, I'm going to keep on the same theme working around a similar idea. What's the sum of the numbers in the square? Too many numbers? Yeah I can see some answers, either we've got some quick calculators going on or people are starting to find other ways to reason to get the answer. This question, this problem itself actually came from a Japanese classroom that I was watching, and I thought this is a really nice one to deal not only with the number but in terms of the explanations. The explanations the children were giving at the time looked at starting with one, moving to a different location, so you get a sense of 'Does it work? Was that just a freaky occurrence up there where it worked or does it work somewhere else? Can I work out the answer again?' And then I'm moving towards thinking about 'Well, how could I explain the answer?' And that's what you can see students were involved in doing in this lesson.
One way they thought about it was saying 'Look, if I think about it in terms of place value, I've got 60 60 60 70 70 70 80 80 80, and I've also got 4 5 6', so there's a sameness going on. Yes the answer is nine times the middle number, the question though is 'Why is it nine times the middle number?' Now you can kind of think well I can look at using some of these number properties that we've got before, and deal with it by breaking it up into two different ways. I've got 60 70 and 80, I've three lots of that, three lots of the other but we also know that 60 70 and 80 is simply three lots of 70, so I've got nine lots of both 70 and nine lots of five, or nine lots of 75. Anyway, the point with that is simply 'How do the students explain their reasoning?' Rather than telling them what reasoning to come up with, and then explaining it for them, because the whole point of trying to get them reasoning around it is before you touch or even get too carried away with the idea of needing to introduce variables and the variable is a [inaudible 00:20:11] to these problems. You can put them in as variables but it's easier to explain the principles of what's going on without introducing xs, ys and zs, I think.
If you liked that one, I'm going to point you across to another one. This is question 41 out of the Red Dragonfly Mathematics Challenge, and the question there is simply 'What would you get if you added up the 81 numbers in the nine times tables, all the way through?' Which is quite a challenging problem in terms of the amount of arithmetic you might need to do if you didn't come up with a good way of finding out the answer. The reasoning there is also about 'Can I use similar types of reasoning in different problems?'
I'm going to continue with that sense of balance around numbers. Because the syllabus also talks about recognising and explaining mathematical relationships using reasoning, we introduce negative numbers as well, and if you look at this particular setup, you've got the total here, and the total here would be ... Yes, no doubt you've guessed it it's zero, because of that sense of symmetry again. You've got that idea here. If you added the same number to each of those terms, you can shift the whole thing into a positive balance point just the same as you've done. Yeah it's a nice web link to the resource as well. What I wanted to do with this though is 'Why is the balance point zero?' And you think well that's nice, that's kind of useful, but I'm dealing with negative numbers. But what else can I do with it? Well, I could use this by rubbing out a bit. Now, how is that useful? Well if I remove that negative six, I'm actually dealing with subtracting a negative number is the same as adding a positive, because you can see how the balance has shifted. There are different ways of dealing with that but I thought just building on the same principle of reasoning around whole numbers, with this we can get to that point of being able to use it also with integers in general.
I said I'd touch briefly on communicating reasoning. One of the things that I talk about is this three step process. Start with getting students to be able to convince themselves that they're right, then to convince a friend, and finally to actually convince an enemy. Somebody who doesn't want to believe you. If you prefer to use the term sceptic, but somebody who doesn't really want to believe you just because you tell them that. Those are the steps we keep going through. 'Okay, convince yourself. Convince a friend. Convince an enemy or a sceptic,' and that may be the teacher sometimes. Typically in stage four we start with questions that require one step of reasoning, progress through to two steps of reasoning. For example, students are expected to use their knowledge of parallel lines and congruent figures to solve numerical exercises on finding unknown lengths and angles and figures. This is standard stage four content, so one step of reasoning, alternate angles probably the quickest way to go with that, well it's the first thing I see. Two steps of reasoning, and you kind of think 'Oh yeah there's a similar kind of diagram I've just got a little more information, little bit more reasoning to go with.'
As well as increasing complexity of chains of reasoning, so there are more steps involved, we actually need to assist students to understand that multiple pathways of reasoning are possible in maths. There is more than one way to get to the answer. How many different ways are there to answer this question? Well, let me think. Yeah I could use co-interior angles. I could use alternate angles, and the angle sum of a straight line. Of course when I'm using co-interior angles, I'm also using adjacent angles. Otherwise? Well, yes I could use the alternate angle from the unknown and bring it inside the triangle and use the angle sum of the triangle.
Now, the interesting thing I like to argue with that is that the syllabus talks about comparing different solutions for the same problem to determine the most efficient method. That's in 5.2. Now with this particular problem, I think that most solutions are of the same level of challenge. I think they're all effectively two steps of reasoning to get to the answer, but I'll leave you to debate that some other time. Not a bad one to have an argument about around the staff room.
It is important for students to understand that there are multiple ways of getting the reasoning out, and to actually be able to see and compare the different ways that occur within a classroom, but 'Recognise that more than one method of the solution is possible' is straight out of the syllabus. If you start with things that are quite reasonable to pick up an [inaudible 00:26:20] convince yourself, the angle sum of a quadrilateral. Have a look at that as a quick example, stage four content, and for a student to convince him or herself, good starting place. Gives them a chance to actually use their protractor skills, something that we often would practise. They can get the sense that they'll convince themselves with a specific case, or you might use the more traditional approach to ripping the corners off and rearranging them, something that I'm sure many of you would do with the triangles and getting the angle sum of triangles demonstrated as being 180, so gathering all the angles at one point.
Of course there's also the idea that you can take a quadrilateral and add in a diagonal and reduce the case to dealing with adjacent angles or the angle sums of two triangles. So what we've got in a normal classroom is the possibility for students to show different types of reasoning. You may have a preference as to which you think is the best, but it depends upon where they've started from. Dealing with the idea of a specific case using your protractor, I'll be fairly happy if most of my students could get the answer to actually add up to 360, because I've seen some of the quadrilaterals they draw at times and it's not hard to be out by a degree or two.
I quite like the idea though of adding in the diagonal, and if I think about it in a different way, because I think that [inaudible 00:28:06] is quite generalizable, you could say that you could deal with ... I think the angles at a point, you could probably keep going for, well, I don't know I'll leave that to you, but I actually like the use of the diagonals. But I want to think about that in just a different way. I want to think about it as in terms of the vertex point. If I talk about where the vertex point, or a point is, I've got one that's sitting over there to divide, and I could teach the angle sum of a quadrilateral or get students to explain it by dealing with it as two triangles. But what if we thought about it a little differently. What if I put the point on the edge. Can I still work out the angle sum of a quadrilateral from the point on the edge? Brief pause, people think about it, I think I get three triangles there but then I have to subtract the straight line, or straight angle BPC. Yep I think I can get there that way.
What about if I take an internal point for the angle sum of a quadrilateral? Think I can do the same thing there, because I often used to use that with a general polygon. I have to subtract the 360 I get at the middle. What about if I put an external point in? The external point P, well I then have to join up that, add in some additional points, and I actually think I've got close to three, four different ways I'm dealing with the angle sum of a quadrilateral. All of them different simply by thinking about where the point ends, or where the point sits, that I deal with. There's a vertex point on an edge, interior, or exterior. Now you might want to spend that amount of time on what you consider to be a small part of the syllabus, the angle sum of a quadrilateral, but it does help to demonstrate that there is more than one way to actually get to the answer. I think that's a useful thing in reasoning.
I'm going to finish on another example which just moves it a little bit higher. Yeah, I think the thing with it is that in terms of confusing most students, that depends on whether or not you provide all the answers. If the answers are actually coming from the students themselves, it's a different thing because then they've got a reason to justify them. Part of it is simply to demonstrate that there is more than one way to get the answer, and that's one of the key things in terms of mathematics. There is more than one path. In this question, I was thinking a little about 'What do you need to do to be able to teach students how to draw lines on diagrams?' And I realised it's one of the hardest things that we actually do, because most of the time we end up drawing in the lines for them, and then get them to solve the problems.
In this question, which I call the angle in the bend, how can you solve that? Well I think the only way to solve that is to make the construction. You can add a line. Why I like this question is that it kind of suggests some good places to add the line. Can add one there, and then I think I'm away because I can use alternate exterior angle sum of a triangle, or I could add it somewhere else. I could add a line through here, parallel. Again, I'm away with that question. But that's not the only place I can add the line. I can also draw in a vertical, and now I'm dealing with perpendiculars between parallel lines, I'm dealing with quadrilaterals, I'm back onto where I was before.
Now, just being mindful of the time I'm going to skip ahead a little bit. That would be one for you to try for yourself. The angle in between parallel lines is a good problem about trying to develop that idea of adding construction lines, which is a very difficult skill to teach. And you can of course use it as you need to within stage four, because demonstrating the angle sum of a triangle is 180 and using that to find the angle sum of a quadrilateral probably works best by the addition of that parallel line. Simply you will get the angle sum of the straight line, transferred through the parallel line properties to get to the total. And you can generalise that process and do the same thing with quadrilateral, via adding all of those angles and then start to think about how they sit together. Then I've got 1+4+5 as 180 degrees, and I'm after 1+2+3+4+5, that gives me 2+3+180 because I've got 1+4+5=180, and 2 and 3 I think are co-interior angles, so we get to the result.
Teaching students to reason relies on students understanding what constitutes a valid argument, appreciating that there can be more than one path to the answer, and recognising you may need to reshape the question to use what you already know. Now I think I've run about four or five minutes past our registered finishing point, so I'm going to leave it at that point. Thank you for your attention and your attendance today. I really just wanted to give you a sense of how could you start to address the key components of reasoning in the syllabus.