January 22, 2024 • 18 mins
In Part 2 of Episode 78 of the NCETM Maths Podcast, Professors Alf Coles and Nathalie Sinclair continue their discussion about the dogmas in maths teaching and learning, with host Julia Thomson. We explore the misconceptions that ‘maths is culture-free’ and that 'maths is for some people and not for others'. We also scrutinise the notion that ‘maths is hard because it is abstract’ and discuss the Concrete, Pictorial, Abstract (CPA) model within mastery and the power of representations in maths.
A transcript (PDF) of this episode is available to download.
Show notes Taking part in the discussion:- Professor Alf Coles, University of Bristol
- Professor Nathalie Sinclair, Simon Fraser University, British Columbia
- Julia Thomson, Communications Manager, NCETM.
- 00:06 - Introduction and welcome
- 00:32 - Exploring the dogma: maths is culture-free
- 05:37 - Maths and climate change
- 07:41 - Debunking the dogma: maths is for some people and not for others
- 12:48 - The power of representations
- 17:02 - Algebra in the primary classroom
- 17:44 - Conclusion and preview for Part 3
- I Can’t Do Maths! Why children say it and how to make a difference by Professor Alf Coles and Professor Nathalie Sinclair (Bloomsbury, 2022)
- NCETM Primary Mastery Professional Development Materials
- Journeys on the Gattegno Tens Chart by Alf Coles, 2014
- Learning Mathematics for an environmentally sustainable future by Karl Bushnell (Association of Teachers of Mathematics, 2018)
- Gattegno’s ‘numbers as lengths’ as mentioned by Alf Coles, Working with Rods and Why (Association of Teachers of Mathematics, 2017)
- I Can't Do Maths - Podcast Episode 78 - Part 1
- Explore previous episodes of the NCETM podcast in our archive.
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Episode Transcript
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(00:30):
Mhm. 6 00:00:32,264.687 --> 00:00:39,774.687 So we now come to the next dogma I found this one really, really interesting, which is that maths is culture free. 7 00:00:40,274.761829932 --> 00:00:48,704.761829932 It was one of the most interesting chapters in the book for me because it just got me thinking so much and looking at things from a different perspective. 8 00:00:48,754.761829932 --> 00:01:05,738.094163266 Can you tell me a little bit more about that dogma? In a way it's fascinating that this is an unusual idea that maths isn't culture free because maths is done by people and people cannot extract themselves from culture when they go into their offices and do their math. 9 00:01:05,998.095163266 --> 00:01:12,388.095163266 So in that sense it's funny, but it's true that we do similar things around the world around mathematics. 10 00:01:12,388.095163266 --> 00:01:16,458.095163266 We share more or less the same number system and talk about shape and so on. 11 00:01:16,458.095163266 --> 00:01:23,48.095163266 So one of the things Alf and I write about is, yeah, it does seem like it's everywhere, sort of more or less the same. 12 00:01:23,48.095163266 --> 00:01:31,988.095163265 And there's certainly what we would call sort of a fantasy that mathematicians have that what they've created is sort of universal. 13 00:01:32,308.095163266 --> 00:01:36,108.095163266 I think people who love math kind of love that idea. 14 00:01:36,108.095163266 --> 00:01:41,578.09516327 And who wouldn't, you know, that you created something that works all the time, everywhere, forever. 15 00:01:41,608.09416327 --> 00:01:43,828.09516327 That would feel very empowering. 16 00:01:44,398.09516327 --> 00:01:49,308.09616327 But that, sort of ignores for example, how mathematics is communicated. 17 00:01:49,308.09616327 --> 00:01:59,108.09616327 So we talk in the book around language and the various ways in which different languages express mathematical ideas that have so many different kinds of connotations. 18 00:01:59,628.09616327 --> 00:02:10,78.09616327 And we talk about gestures as well, which is another aspect of communication and how these are very different across cultures and can be mobilized in different ways. 19 00:02:10,708.09616327 --> 00:02:18,38.09516327 And we also talk about how mathematical ideas have ideological and aesthetic connections to them. 20 00:02:18,38.09616327 --> 00:02:28,208.09616327 So there is no real world against which we can say, oh, this is a good mathematical idea, and this is a bad mathematical idea, because it's all sort of made up. 21 00:02:28,848.09616327 --> 00:02:34,481.4294966 And so there has to be choice that comes along of people saying, Okay, this one is interesting. 22 00:02:34,741.4294966 --> 00:02:37,921.4294966 Let's put it in the curriculum and make everybody learn it right. 23 00:02:37,928.09616327 --> 00:02:50,885.58695918 So what are those choices that get made? It's sort of similar to why are we putting the Mona Lisa in the museum? What is it about that that we all find so great that we think everybody should see it. 24 00:02:51,715.58695918 --> 00:03:13,512.25412585 So I think one of the ways in which these aspects of culture can be made more available to students would be to think about definitions, for example, if you're in a grade three class and you're talking with students about, how would you define a square and different students can define it in different ways, four equal sides they could talk about it's something that's symmetric. 25 00:03:13,522.25412585 --> 00:03:18,102.25412585 If you turn it around by 90 degrees, four times lots of different ways. 26 00:03:18,322.25412585 --> 00:03:24,112.32079252 You could talk about the diagonals that intersect at 90 degrees and bisect each other. 27 00:03:24,552.32079252 --> 00:03:40,45.65412585 All of these different ways are ways of defining squares and the students could discuss, okay, well, which one is better and maybe some will be more efficient, some will be more understandable, some will be more visual, which will appeal to certain people over others. 28 00:03:40,505.65412585 --> 00:03:50,125.65312585 And all of this discussion about what a definition of a square is will bring out these different aesthetic aspects of which ones do we choose. 29 00:03:50,155.65212585 --> 00:04:06,785.65312585 Then they could go look at definitions of squares and textbooks around the world and see actually, people in Canada define squares differently and people in the US and Australia and some countries have inclusive definitions of squares and some have exclusive definitions of squares. 30 00:04:07,65.65312585 --> 00:04:09,5.65312585 So it's not the same thing everywhere. 31 00:04:09,345.65212585 --> 00:04:15,275.65312585 And the reason why people choose one over the other depends on some of their preferences. 32 00:04:15,275.75312585 --> 00:04:26,642.01693764 Some people really like to have definitions that will include all of the shapes that are contained within the square because that makes it easier to prove things. 33 00:04:26,642.01693764 --> 00:04:35,440.56795805 You don't have to do it over and over again for each shape and some prefer to have a definition of a square that excludes other things that are not quite square. 34 00:04:36,90.56795805 --> 00:04:52,710.56795805 So I think definitions are a fruitful place to help students see how much choice is involved and how much those choices are based on certain preferences that we can see even a very simple idea of, of defining a square. 35 00:04:54,277.23462472 --> 00:05:08,616.01796987 I was interested particularly in that because maths is done by humans and humans have their own sets of values and their own backgrounds and their own levels of wealth or education or, you know, they may be in the West or not. 36 00:05:09,246.01796987 --> 00:05:13,146.01796987 That how we do maths is also a cultural thing. 37 00:05:13,476.01696987 --> 00:05:21,249.35030321 I was quite interested in how that might make maths feel more relevant to some students who were doing it, maybe more personal. 38 00:05:21,370.39916216 --> 00:05:25,800.16800908 Maths can feel very impersonal sometimes, the way it's taught can be very dry. 39 00:05:26,420.16800908 --> 00:05:40,923.39984241 And one of the threads running through your book really was the relational aspect of maths You were looking at an activity by Carl Bushnell, who was looking at maths problems in relation to climate Yeah, Carl went through his teacher education at Bristol. 40 00:05:40,963.39984241 --> 00:05:50,843.39984241 I think it's really, really interesting, you know, he's continued to do bits of writing around how questions around the climate, for instance, might be relevant to look to the mathematics classroom. 41 00:05:50,973.39984241 --> 00:05:53,373.39984241 And with some really, really interesting things that he's doing. 42 00:05:53,383.39984241 --> 00:06:03,300.06600908 So, in one of the examples, he takes students through a set of problems where you end up working out how much would the sea level rise if the whole of the Greenland ice sheet melted. 43 00:06:03,400.06600908 --> 00:06:04,950.06600908 And it's really pretty straightforward. 44 00:06:05,130.06600908 --> 00:06:10,595.06600908 You need to know some key bits of information, but actually the mathematics involved is pretty straightforward. 45 00:06:10,605.06600908 --> 00:06:12,445.06600908 It's nothing more than arithmetic, really. 46 00:06:12,845.06600908 --> 00:06:42,401.99828608 One of the things Carl actually writes about, at what point do our responsibilities as math teachers stop? I mean, it feels like when we come to conclusions that actually are beginning to have existential implications on Earth, that actually it's hard to contain that what we do in maths is just to do with numbers that actually we've got to start thinking about the implications of what this means and even just to allow space for the expression of anxiety perhaps or sort of horror or you know hopes. 47 00:06:43,411.99828608 --> 00:07:20,246.99928608 I mean I do think that we have a responsibility to also find places of hope for students and I think that all links in with the idea that maths is not culture free, that I think both of us would feel the mathematics doesn't stop at the point of getting the answer to the number of meters the sea level rises, that, elements of mathematics are also around what's that going to mean for different people around the world? I can imagine that looking at maths in that way really appealing to secondary and post 16 students, particularly maths not being something that children or young people are going to need in the future. 48 00:07:20,276.99928608 --> 00:07:21,796.99928608 You know, I'm never going to do this again. 49 00:07:22,136.99928608 --> 00:07:40,591.61153098 When actually it's something that is incredibly important and particularly young people are so passionate about the climate and about what's going on in the world, I think it would probably surprise them to know how important maths is in tackling some of those problems. 50 00:07:41,451.61153098 --> 00:07:46,681.51153098 Moving on to Dogma D, which is the maths myth that the NCETM is keen to dispel. 51 00:07:46,711.51153098 --> 00:07:49,301.51153098 Maths is for some people and not for others. 52 00:07:49,791.51153098 --> 00:08:05,565.66309348 And in this chapter, we're thinking a little bit more about setting and differentiation and that sort of thing, and how those children can get, sometimes really young, that idea that they're not good at maths because of the way that adults have organized them in the classroom. 53 00:08:05,565.66309348 --> 00:08:14,585.76309348 I'm interested to know, and I think that teachers will be interested and parents will be interested to know how, we might go about tackling that particular dogma. 54 00:08:16,275.76309348 --> 00:08:16,635.76309348 Okay. 55 00:08:16,695.76309348 --> 00:08:17,625.76309348 So yes. 56 00:08:17,735.76309348 --> 00:08:20,655.76309348 So this dogma that maths is for some people and not for others. 57 00:08:20,655.76309348 --> 00:08:30,285.76309348 I mean, I think we both felt this was a really significant one within the book for all the kind of social justice reasons that I think you're alluding to in the way you're talking about it. 58 00:08:30,285.76309348 --> 00:08:49,260.76209348 And again, yeah, I mean, I think for me, this is one of the really, really significant things that the NCETM has been trying to push and I mean, I'm personally delighted to see that mixed attainment teaching seems to be on the rise in England, I think, particularly at primary, but but also as far as I can tell at secondary school as well. 59 00:08:49,760.76209348 --> 00:09:02,982.4572386 I mean, it's interesting that one of the ideas that's sometimes used to help think about 'maths is for some people not for others', or one of the ideas that's used to combat this, is the idea of growth and fixed mindsets. 60 00:09:03,32.4572386 --> 00:09:15,102.4572386 And while we see some interesting things here, I think we both have a worry that the idea of mindsets brings us back to the idea that maths and doing well at school is all about individual characteristics. 61 00:09:15,352.4572386 --> 00:09:24,562.45673861 And one of the things I think is that it's not made clear in the idea of growth and fixed mindsets is really quite how you move from a fixed mindset to a growth mindset. 62 00:09:24,642.45673861 --> 00:09:37,992.45673861 So I think perhaps one of the things that's happened is that, whereas in the past I might've talked about these students are in the bottom set or these students have got low maths ability or then more recently, hopefully, these students have got low maths attainment, low prior attainment. 63 00:09:38,332.45673861 --> 00:09:41,742.45773861 So now I might think, these students have all got fixed mindset. 64 00:09:41,862.45773861 --> 00:09:49,112.5582386 But it has this similar kind of feel really that it's like the problem's in the students and I've got to find some way of getting them to shift. 65 00:09:49,222.5582386 --> 00:10:05,792.5582386 And I think, we're really proposing in the book that if a child you're teaching has taught themselves a language, their first language, then really they have all the skills they need to succeed at school level mathematics way beyond primary school. 66 00:10:06,122.5582386 --> 00:10:10,802.5582386 And so if they're not succeeding, there is actually no deficit in themselves. 67 00:10:11,72.5582386 --> 00:10:14,182.5582386 That there cannot be because they because they taught themselves to speak. 68 00:10:14,302.5582386 --> 00:10:17,452.5582386 So the issue is how they've come to relate to mathematics. 69 00:10:17,612.5582386 --> 00:10:35,45.99107193 So I think what we're trying to suggest is that we encourage a framing that all learners who can speak a language are incredibly powerful learners and if we recognize that achievement, then the thing is how can we build our learning in our classrooms on the basis of kind of respect for that incredible learning feat that they have achieved. 70 00:10:35,205.99107193 --> 00:10:40,629.14307194 And I suppose it's that kind of idea that we're trying to move towards in the book and suggest some strategies for. 71 00:10:40,629.14307194 --> 00:10:47,808.08751638 Just to add on to what Alf was saying and to keep the idea of these being dogmas so that there's like they come from somewhere. 72 00:10:47,808.08751638 --> 00:10:55,669.45939846 I think in some ways in which we have historically taught mathematics, it has actually been only for some people. 73 00:10:56,359.45939846 --> 00:11:13,429.00134858 Right from the beginning, schools were for rich males in general, and then it took a long time for more and more kids to be welcome at school, but certain ways of keeping kids in lines and seated at their desks with their hands behind their back just doesn't work for a lot of people. 74 00:11:13,859.00234858 --> 00:11:29,586.84561842 And so it's not surprising if people say, Oh, math is not for me or that I'm bad at math because what they're really saying is I'm bad at learning math, if that's how it's going to be presented to me, right? It's kind of like going back to Alf's analogy of learning language. 75 00:11:29,586.84561842 --> 00:11:36,906.84561842 It's like if you were just learning a language from one person who was giving you, you know, five words every day and that was it. 76 00:11:37,316.84561842 --> 00:11:40,116.84561842 And you had to master them before you got the next five words. 77 00:11:40,416.84561842 --> 00:12:00,41.84661842 You probably wouldn't think you would be very good at learning language too, but actually, you know, most kids are involved in a lot of complex, diverse experiences in which they pick up ways of speaking in different ways over time and aren't hit over the hands if they get the wrong word often. 78 00:12:00,101.84661842 --> 00:12:17,626.85711842 So I think one of the things we were really pointing to is the importance, for example, of bringing in visual ways of understanding mathematics or embodied ways of understanding mathematics so that doesn't always have to be this one track, which is highly symbolic, let's say. 79 00:12:17,816.85811842 --> 00:12:37,17.79839393 Not that there's anything wrong with with symbols, but some people just have a much easier way of feeling comfortable with the mathematical ideas if they're given options into how they're experiencing them so I just wanted to say that the dogma has some basis in real experiences that people have had. 80 00:12:37,817.79839393 --> 00:12:41,317.79839393 And it probably does come from that negative association. 81 00:12:41,357.79839393 --> 00:12:43,937.79739393 So if you're not enjoying it, then it's not for me. 82 00:12:43,947.79839393 --> 00:12:47,997.69939393 And I think, that's fairly understandable, definitely. 83 00:12:48,497.69939393 --> 00:12:53,227.69939393 So our next dogma is that maths is hard because it's abstract. 84 00:12:53,727.69939393 --> 00:12:58,547.69939393 Which again, I thought was a really interesting chapter because it didn't quite go where I was expecting it to. 85 00:12:58,927.69939393 --> 00:13:00,807.69839393 And I found it really fascinating. 86 00:13:01,151.03172726 --> 00:13:05,881.03172726 I was thinking of concrete as being purely representational, but you explored sort of different. 87 00:13:06,297.69939393 --> 00:13:08,748.57921252 views of the concrete and the abstract. 88 00:13:09,208.57921252 --> 00:13:12,818.68352091 So I'm interested to find out a little bit more about this dogma. 89 00:13:13,68.68352091 --> 00:13:15,218.68352091 Maths is hard because it is abstract. 90 00:13:15,238.68352091 --> 00:13:21,98.68352091 Yeah, I mean, we do play around a bit in the chapter with what those words might mean but I think I won't go into that now. 91 00:13:21,98.68352091 --> 00:14:01,257.08988187 If we just stay with perhaps a more typical meaning of the word abstract there are elements of maths, which do fit that, which do seem to be abstract and if you read a page of symbols, you know, what on earth does it seem to be about? But I think one of the things we're keen to point to is that, children actually have a lot of skills in abstract thinking and again, to come back to language, even if you think of a word like chair that's a pretty arbitrary symbol for this collection of objects, all a little bit different other, but share some similarities well, really, what's that if that's not abstract that it seems to be language is inherently abstract. 92 00:14:01,287.08988187 --> 00:14:13,377.08888187 So, whatever difficulties children might be having with mathematics, it can't be because they've got any deficits with abstract thinking so that would be one reading of what we're trying to say in this chapter. 93 00:14:14,457.08788187 --> 00:14:26,266.98988187 I mean, I think another thing that we want to suggest is that the idea that there has to be this movement from concrete to pictorial to abstract where you try and get quickly to the abstract and then stay there. 94 00:14:26,506.98988187 --> 00:14:28,66.99088187 I don't think is very helpful. 95 00:14:28,126.99088187 --> 00:14:47,326.98988187 I think Singapore is one of the places where this model is used very extensively, and I think it would be fair to say that actually what happens in Singapore is, as quickly as possible, you get these three different ways of thinking, but actually you work on the links and connections between them, and all three of them stay around really for quite a long time. 96 00:14:47,386.98988187 --> 00:14:58,603.65504853 One of the things that I really love in the NCETM ideas is the idea of having a few representations that keep coming back through through the curriculum from primary into secondary. 97 00:14:58,933.65604853 --> 00:15:00,793.65604853 And so one of those would be the number line. 98 00:15:00,893.65604853 --> 00:15:03,463.65554853 Now we might see that as a pictorial representation. 99 00:15:03,463.65554853 --> 00:15:04,233.65554853 I'm not sure. 100 00:15:04,813.65554853 --> 00:15:10,688.65654853 Why would you ever want to not use a number line? I mean, I still use a number line if I'm thinking about it. 101 00:15:10,998.65654853 --> 00:15:14,68.65654853 So there isn't this sense that I've got to somehow go somewhere else before that. 102 00:15:14,228.65654853 --> 00:15:20,218.65654853 If I choose representations that are powerful enough that they can stay with me throughout my mathematical career. 103 00:15:20,348.65654853 --> 00:15:27,708.65654853 At secondary level, again, the NCETM endorses trigonometry taught through a circle and a circle image. 104 00:15:27,788.65654853 --> 00:15:33,271.44226282 Again, that's a representation that you never need to let go of, in order to work with trigonometry. 105 00:15:33,321.44226282 --> 00:15:35,661.44226282 And I suppose just may, maybe one other brief example. 106 00:15:35,661.44226282 --> 00:15:38,13.10892948 I think Gattegno who you mentioned. 107 00:15:38,73.10892948 --> 00:15:46,923.10892948 In his curriculum, he suggests using number as length, thinking about numbers as lengths, which is a sort of way of making them concrete. 108 00:15:47,33.10892948 --> 00:15:50,752.10892948 And, again, it's actually a way of thinking about number that you never have to let go of. 109 00:15:50,752.10892948 --> 00:15:55,402.10892948 It's an entirely consistent way of thinking about number right until higher levels of mathematics. 110 00:15:55,512.10892948 --> 00:16:12,742.10792948 And again, within the NCETM materials I think one really fantastic innovation is that alongside introducing numbers as objects there is a strand through the professional development materials of dealing with number as length and so number as object number as length run alongside. 111 00:16:12,772.10692948 --> 00:16:21,352.10792948 And I think that there's a real power there in terms of offering students different ways into number and making links and connections across these two representations. 112 00:16:21,718.67459615 --> 00:16:30,198.77459615 So numbers as length, you might be thinking of something like Cuisenaire rods or bar model type representations. 113 00:16:30,318.77459615 --> 00:16:30,508.77459615 Yeah. 114 00:16:30,538.77459615 --> 00:16:31,408.77359615 It blew my mind... 115 00:16:31,458.77459615 --> 00:16:33,188.77359615 I once solved a problem. 116 00:16:33,578.77459615 --> 00:16:36,228.77359615 It was a GCSE problem and I was just playing around. 117 00:16:36,228.77359615 --> 00:16:37,118.77359615 It was on a podcast. 118 00:16:37,613.77459615 --> 00:16:39,63.77459615 And I solved it with a bar model. 119 00:16:39,503.77459615 --> 00:16:42,323.7745961 And then the hosts were talking about how they solved it. 120 00:16:42,323.7745961 --> 00:16:44,933.7745961 And one of them used a bar model and the other one used algebra. 121 00:16:45,183.7745961 --> 00:16:47,853.7745961 And I thought, bar model is algebra. 122 00:16:48,403.7735961 --> 00:17:02,253.7725961 How is that? I had no idea that that's actually what you're doing when you're, when you're solving some of these sort of, primary school problems with the bar model, it's actually algebra, which is just like, but there you go, complexity. 123 00:17:03,23.7735961 --> 00:17:15,83.7735961 I mean, in Gattegno's original curriculum he proposes you teach algebra before arithmetic, that you work on the sort of more abstract relationships between lengths and how lengths fit together. 124 00:17:15,173.7735961 --> 00:17:17,723.7735961 And only later do you then put, put numbers onto it. 125 00:17:18,603.7735961 --> 00:17:29,93.7735961 I remember when I was teaching in Year 6 and the children would know that they were going to be doing algebra and they'd say, when are we going to be doing algebra? Is it really hard? I can't wait to do algebra. 126 00:17:29,93.7735961 --> 00:17:39,772.8620315 And obviously they were terrified, I've just thought now that some of the problems we were already solving I would have loved to have been able to turn around to them and said, you did it last year in Year 5, you know, that was algebra. 127 00:17:41,92.8620315 --> 00:17:42,182.8620315 Missed opportunity. 128 00:17:44,772.8244139 --> 00:17:47,2.8244139 And that brings us to the end of part two. 129 00:17:47,652.8244139 --> 00:17:53,322.8244139 I hope our discussion of Alf and Nathalie's really fascinating and thought provoking book has left you wanting to read it. 130 00:17:53,852.8244139 --> 00:18:00,972.8244139 I think anyone interested in how children learn or how we might go about teaching mathematics successfully will find it such a fascinating read. 131 00:18:01,902.8244139 --> 00:18:09,602.8244139 Do come back and join us for the final third part of our conversation where we put your questions, shared with us on social media, to Alf and Nathalie. 132 00:18:10,142.8244139 --> 00:18:17,812.8244139 And in the meantime, we'd love it if you could share this episode with colleagues, like the podcast and subscribe to our channel wherever you get your podcasts. 133 00:18:18,572.8244139 --> 00:18:23,302.8244139 You can also hit the notification bell to make sure you are notified when the next episode drops. 134 00:18:23,772.8244139 --> 00:18:28,132.8244139 And finally, if you're on Instagram, do follow us at @themathspodcast. 135 00:18:28,652.8234139 --> 00:18:29,612.8244139 Thanks for listening.
Mhm. 6 00:00:32,264.687 --> 00:00:39,774.687 So we now come to the next dogma I found this one really, really interesting, which is that maths is culture free. 7 00:00:40,274.761829932 --> 00:00:48,704.761829932 It was one of the most interesting chapters in the book for me because it just got me thinking so much and looking at things from a different perspective. 8 00:00:48,754.761829932 --> 00:01:05,738.094163266 Can you tell me a little bit more about that dogma? In a way it's fascinating that this is an unusual idea that maths isn't culture free because maths is done by people and people cannot extract themselves from culture when they go into their offices and do their math. 9 00:01:05,998.095163266 --> 00:01:12,388.095163266 So in that sense it's funny, but it's true that we do similar things around the world around mathematics. 10 00:01:12,388.095163266 --> 00:01:16,458.095163266 We share more or less the same number system and talk about shape and so on. 11 00:01:16,458.095163266 --> 00:01:23,48.095163266 So one of the things Alf and I write about is, yeah, it does seem like it's everywhere, sort of more or less the same. 12 00:01:23,48.095163266 --> 00:01:31,988.095163265 And there's certainly what we would call sort of a fantasy that mathematicians have that what they've created is sort of universal. 13 00:01:32,308.095163266 --> 00:01:36,108.095163266 I think people who love math kind of love that idea. 14 00:01:36,108.095163266 --> 00:01:41,578.09516327 And who wouldn't, you know, that you created something that works all the time, everywhere, forever. 15 00:01:41,608.09416327 --> 00:01:43,828.09516327 That would feel very empowering. 16 00:01:44,398.09516327 --> 00:01:49,308.09616327 But that, sort of ignores for example, how mathematics is communicated. 17 00:01:49,308.09616327 --> 00:01:59,108.09616327 So we talk in the book around language and the various ways in which different languages express mathematical ideas that have so many different kinds of connotations. 18 00:01:59,628.09616327 --> 00:02:10,78.09616327 And we talk about gestures as well, which is another aspect of communication and how these are very different across cultures and can be mobilized in different ways. 19 00:02:10,708.09616327 --> 00:02:18,38.09516327 And we also talk about how mathematical ideas have ideological and aesthetic connections to them. 20 00:02:18,38.09616327 --> 00:02:28,208.09616327 So there is no real world against which we can say, oh, this is a good mathematical idea, and this is a bad mathematical idea, because it's all sort of made up. 21 00:02:28,848.09616327 --> 00:02:34,481.4294966 And so there has to be choice that comes along of people saying, Okay, this one is interesting. 22 00:02:34,741.4294966 --> 00:02:37,921.4294966 Let's put it in the curriculum and make everybody learn it right. 23 00:02:37,928.09616327 --> 00:02:50,885.58695918 So what are those choices that get made? It's sort of similar to why are we putting the Mona Lisa in the museum? What is it about that that we all find so great that we think everybody should see it. 24 00:02:51,715.58695918 --> 00:03:13,512.25412585 So I think one of the ways in which these aspects of culture can be made more available to students would be to think about definitions, for example, if you're in a grade three class and you're talking with students about, how would you define a square and different students can define it in different ways, four equal sides they could talk about it's something that's symmetric. 25 00:03:13,522.25412585 --> 00:03:18,102.25412585 If you turn it around by 90 degrees, four times lots of different ways. 26 00:03:18,322.25412585 --> 00:03:24,112.32079252 You could talk about the diagonals that intersect at 90 degrees and bisect each other. 27 00:03:24,552.32079252 --> 00:03:40,45.65412585 All of these different ways are ways of defining squares and the students could discuss, okay, well, which one is better and maybe some will be more efficient, some will be more understandable, some will be more visual, which will appeal to certain people over others. 28 00:03:40,505.65412585 --> 00:03:50,125.65312585 And all of this discussion about what a definition of a square is will bring out these different aesthetic aspects of which ones do we choose. 29 00:03:50,155.65212585 --> 00:04:06,785.65312585 Then they could go look at definitions of squares and textbooks around the world and see actually, people in Canada define squares differently and people in the US and Australia and some countries have inclusive definitions of squares and some have exclusive definitions of squares. 30 00:04:07,65.65312585 --> 00:04:09,5.65312585 So it's not the same thing everywhere. 31 00:04:09,345.65212585 --> 00:04:15,275.65312585 And the reason why people choose one over the other depends on some of their preferences. 32 00:04:15,275.75312585 --> 00:04:26,642.01693764 Some people really like to have definitions that will include all of the shapes that are contained within the square because that makes it easier to prove things. 33 00:04:26,642.01693764 --> 00:04:35,440.56795805 You don't have to do it over and over again for each shape and some prefer to have a definition of a square that excludes other things that are not quite square. 34 00:04:36,90.56795805 --> 00:04:52,710.56795805 So I think definitions are a fruitful place to help students see how much choice is involved and how much those choices are based on certain preferences that we can see even a very simple idea of, of defining a square. 35 00:04:54,277.23462472 --> 00:05:08,616.01796987 I was interested particularly in that because maths is done by humans and humans have their own sets of values and their own backgrounds and their own levels of wealth or education or, you know, they may be in the West or not. 36 00:05:09,246.01796987 --> 00:05:13,146.01796987 That how we do maths is also a cultural thing. 37 00:05:13,476.01696987 --> 00:05:21,249.35030321 I was quite interested in how that might make maths feel more relevant to some students who were doing it, maybe more personal. 38 00:05:21,370.39916216 --> 00:05:25,800.16800908 Maths can feel very impersonal sometimes, the way it's taught can be very dry. 39 00:05:26,420.16800908 --> 00:05:40,923.39984241 And one of the threads running through your book really was the relational aspect of maths You were looking at an activity by Carl Bushnell, who was looking at maths problems in relation to climate Yeah, Carl went through his teacher education at Bristol. 40 00:05:40,963.39984241 --> 00:05:50,843.39984241 I think it's really, really interesting, you know, he's continued to do bits of writing around how questions around the climate, for instance, might be relevant to look to the mathematics classroom. 41 00:05:50,973.39984241 --> 00:05:53,373.39984241 And with some really, really interesting things that he's doing. 42 00:05:53,383.39984241 --> 00:06:03,300.06600908 So, in one of the examples, he takes students through a set of problems where you end up working out how much would the sea level rise if the whole of the Greenland ice sheet melted. 43 00:06:03,400.06600908 --> 00:06:04,950.06600908 And it's really pretty straightforward. 44 00:06:05,130.06600908 --> 00:06:10,595.06600908 You need to know some key bits of information, but actually the mathematics involved is pretty straightforward. 45 00:06:10,605.06600908 --> 00:06:12,445.06600908 It's nothing more than arithmetic, really. 46 00:06:12,845.06600908 --> 00:06:42,401.99828608 One of the things Carl actually writes about, at what point do our responsibilities as math teachers stop? I mean, it feels like when we come to conclusions that actually are beginning to have existential implications on Earth, that actually it's hard to contain that what we do in maths is just to do with numbers that actually we've got to start thinking about the implications of what this means and even just to allow space for the expression of anxiety perhaps or sort of horror or you know hopes. 47 00:06:43,411.99828608 --> 00:07:20,246.99928608 I mean I do think that we have a responsibility to also find places of hope for students and I think that all links in with the idea that maths is not culture free, that I think both of us would feel the mathematics doesn't stop at the point of getting the answer to the number of meters the sea level rises, that, elements of mathematics are also around what's that going to mean for different people around the world? I can imagine that looking at maths in that way really appealing to secondary and post 16 students, particularly maths not being something that children or young people are going to need in the future. 48 00:07:20,276.99928608 --> 00:07:21,796.99928608 You know, I'm never going to do this again. 49 00:07:22,136.99928608 --> 00:07:40,591.61153098 When actually it's something that is incredibly important and particularly young people are so passionate about the climate and about what's going on in the world, I think it would probably surprise them to know how important maths is in tackling some of those problems. 50 00:07:41,451.61153098 --> 00:07:46,681.51153098 Moving on to Dogma D, which is the maths myth that the NCETM is keen to dispel. 51 00:07:46,711.51153098 --> 00:07:49,301.51153098 Maths is for some people and not for others. 52 00:07:49,791.51153098 --> 00:08:05,565.66309348 And in this chapter, we're thinking a little bit more about setting and differentiation and that sort of thing, and how those children can get, sometimes really young, that idea that they're not good at maths because of the way that adults have organized them in the classroom. 53 00:08:05,565.66309348 --> 00:08:14,585.76309348 I'm interested to know, and I think that teachers will be interested and parents will be interested to know how, we might go about tackling that particular dogma. 54 00:08:16,275.76309348 --> 00:08:16,635.76309348 Okay. 55 00:08:16,695.76309348 --> 00:08:17,625.76309348 So yes. 56 00:08:17,735.76309348 --> 00:08:20,655.76309348 So this dogma that maths is for some people and not for others. 57 00:08:20,655.76309348 --> 00:08:30,285.76309348 I mean, I think we both felt this was a really significant one within the book for all the kind of social justice reasons that I think you're alluding to in the way you're talking about it. 58 00:08:30,285.76309348 --> 00:08:49,260.76209348 And again, yeah, I mean, I think for me, this is one of the really, really significant things that the NCETM has been trying to push and I mean, I'm personally delighted to see that mixed attainment teaching seems to be on the rise in England, I think, particularly at primary, but but also as far as I can tell at secondary school as well. 59 00:08:49,760.76209348 --> 00:09:02,982.4572386 I mean, it's interesting that one of the ideas that's sometimes used to help think about 'maths is for some people not for others', or one of the ideas that's used to combat this, is the idea of growth and fixed mindsets. 60 00:09:03,32.4572386 --> 00:09:15,102.4572386 And while we see some interesting things here, I think we both have a worry that the idea of mindsets brings us back to the idea that maths and doing well at school is all about individual characteristics. 61 00:09:15,352.4572386 --> 00:09:24,562.45673861 And one of the things I think is that it's not made clear in the idea of growth and fixed mindsets is really quite how you move from a fixed mindset to a growth mindset. 62 00:09:24,642.45673861 --> 00:09:37,992.45673861 So I think perhaps one of the things that's happened is that, whereas in the past I might've talked about these students are in the bottom set or these students have got low maths ability or then more recently, hopefully, these students have got low maths attainment, low prior attainment. 63 00:09:38,332.45673861 --> 00:09:41,742.45773861 So now I might think, these students have all got fixed mindset. 64 00:09:41,862.45773861 --> 00:09:49,112.5582386 But it has this similar kind of feel really that it's like the problem's in the students and I've got to find some way of getting them to shift. 65 00:09:49,222.5582386 --> 00:10:05,792.5582386 And I think, we're really proposing in the book that if a child you're teaching has taught themselves a language, their first language, then really they have all the skills they need to succeed at school level mathematics way beyond primary school. 66 00:10:06,122.5582386 --> 00:10:10,802.5582386 And so if they're not succeeding, there is actually no deficit in themselves. 67 00:10:11,72.5582386 --> 00:10:14,182.5582386 That there cannot be because they because they taught themselves to speak. 68 00:10:14,302.5582386 --> 00:10:17,452.5582386 So the issue is how they've come to relate to mathematics. 69 00:10:17,612.5582386 --> 00:10:35,45.99107193 So I think what we're trying to suggest is that we encourage a framing that all learners who can speak a language are incredibly powerful learners and if we recognize that achievement, then the thing is how can we build our learning in our classrooms on the basis of kind of respect for that incredible learning feat that they have achieved. 70 00:10:35,205.99107193 --> 00:10:40,629.14307194 And I suppose it's that kind of idea that we're trying to move towards in the book and suggest some strategies for. 71 00:10:40,629.14307194 --> 00:10:47,808.08751638 Just to add on to what Alf was saying and to keep the idea of these being dogmas so that there's like they come from somewhere. 72 00:10:47,808.08751638 --> 00:10:55,669.45939846 I think in some ways in which we have historically taught mathematics, it has actually been only for some people. 73 00:10:56,359.45939846 --> 00:11:13,429.00134858 Right from the beginning, schools were for rich males in general, and then it took a long time for more and more kids to be welcome at school, but certain ways of keeping kids in lines and seated at their desks with their hands behind their back just doesn't work for a lot of people. 74 00:11:13,859.00234858 --> 00:11:29,586.84561842 And so it's not surprising if people say, Oh, math is not for me or that I'm bad at math because what they're really saying is I'm bad at learning math, if that's how it's going to be presented to me, right? It's kind of like going back to Alf's analogy of learning language. 75 00:11:29,586.84561842 --> 00:11:36,906.84561842 It's like if you were just learning a language from one person who was giving you, you know, five words every day and that was it. 76 00:11:37,316.84561842 --> 00:11:40,116.84561842 And you had to master them before you got the next five words. 77 00:11:40,416.84561842 --> 00:12:00,41.84661842 You probably wouldn't think you would be very good at learning language too, but actually, you know, most kids are involved in a lot of complex, diverse experiences in which they pick up ways of speaking in different ways over time and aren't hit over the hands if they get the wrong word often. 78 00:12:00,101.84661842 --> 00:12:17,626.85711842 So I think one of the things we were really pointing to is the importance, for example, of bringing in visual ways of understanding mathematics or embodied ways of understanding mathematics so that doesn't always have to be this one track, which is highly symbolic, let's say. 79 00:12:17,816.85811842 --> 00:12:37,17.79839393 Not that there's anything wrong with with symbols, but some people just have a much easier way of feeling comfortable with the mathematical ideas if they're given options into how they're experiencing them so I just wanted to say that the dogma has some basis in real experiences that people have had. 80 00:12:37,817.79839393 --> 00:12:41,317.79839393 And it probably does come from that negative association. 81 00:12:41,357.79839393 --> 00:12:43,937.79739393 So if you're not enjoying it, then it's not for me. 82 00:12:43,947.79839393 --> 00:12:47,997.69939393 And I think, that's fairly understandable, definitely. 83 00:12:48,497.69939393 --> 00:12:53,227.69939393 So our next dogma is that maths is hard because it's abstract. 84 00:12:53,727.69939393 --> 00:12:58,547.69939393 Which again, I thought was a really interesting chapter because it didn't quite go where I was expecting it to. 85 00:12:58,927.69939393 --> 00:13:00,807.69839393 And I found it really fascinating. 86 00:13:01,151.03172726 --> 00:13:05,881.03172726 I was thinking of concrete as being purely representational, but you explored sort of different. 87 00:13:06,297.69939393 --> 00:13:08,748.57921252 views of the concrete and the abstract. 88 00:13:09,208.57921252 --> 00:13:12,818.68352091 So I'm interested to find out a little bit more about this dogma. 89 00:13:13,68.68352091 --> 00:13:15,218.68352091 Maths is hard because it is abstract. 90 00:13:15,238.68352091 --> 00:13:21,98.68352091 Yeah, I mean, we do play around a bit in the chapter with what those words might mean but I think I won't go into that now. 91 00:13:21,98.68352091 --> 00:14:01,257.08988187 If we just stay with perhaps a more typical meaning of the word abstract there are elements of maths, which do fit that, which do seem to be abstract and if you read a page of symbols, you know, what on earth does it seem to be about? But I think one of the things we're keen to point to is that, children actually have a lot of skills in abstract thinking and again, to come back to language, even if you think of a word like chair that's a pretty arbitrary symbol for this collection of objects, all a little bit different other, but share some similarities well, really, what's that if that's not abstract that it seems to be language is inherently abstract. 92 00:14:01,287.08988187 --> 00:14:13,377.08888187 So, whatever difficulties children might be having with mathematics, it can't be because they've got any deficits with abstract thinking so that would be one reading of what we're trying to say in this chapter. 93 00:14:14,457.08788187 --> 00:14:26,266.98988187 I mean, I think another thing that we want to suggest is that the idea that there has to be this movement from concrete to pictorial to abstract where you try and get quickly to the abstract and then stay there. 94 00:14:26,506.98988187 --> 00:14:28,66.99088187 I don't think is very helpful. 95 00:14:28,126.99088187 --> 00:14:47,326.98988187 I think Singapore is one of the places where this model is used very extensively, and I think it would be fair to say that actually what happens in Singapore is, as quickly as possible, you get these three different ways of thinking, but actually you work on the links and connections between them, and all three of them stay around really for quite a long time. 96 00:14:47,386.98988187 --> 00:14:58,603.65504853 One of the things that I really love in the NCETM ideas is the idea of having a few representations that keep coming back through through the curriculum from primary into secondary. 97 00:14:58,933.65604853 --> 00:15:00,793.65604853 And so one of those would be the number line. 98 00:15:00,893.65604853 --> 00:15:03,463.65554853 Now we might see that as a pictorial representation. 99 00:15:03,463.65554853 --> 00:15:04,233.65554853 I'm not sure. 100 00:15:04,813.65554853 --> 00:15:10,688.65654853 Why would you ever want to not use a number line? I mean, I still use a number line if I'm thinking about it. 101 00:15:10,998.65654853 --> 00:15:14,68.65654853 So there isn't this sense that I've got to somehow go somewhere else before that. 102 00:15:14,228.65654853 --> 00:15:20,218.65654853 If I choose representations that are powerful enough that they can stay with me throughout my mathematical career. 103 00:15:20,348.65654853 --> 00:15:27,708.65654853 At secondary level, again, the NCETM endorses trigonometry taught through a circle and a circle image. 104 00:15:27,788.65654853 --> 00:15:33,271.44226282 Again, that's a representation that you never need to let go of, in order to work with trigonometry. 105 00:15:33,321.44226282 --> 00:15:35,661.44226282 And I suppose just may, maybe one other brief example. 106 00:15:35,661.44226282 --> 00:15:38,13.10892948 I think Gattegno who you mentioned. 107 00:15:38,73.10892948 --> 00:15:46,923.10892948 In his curriculum, he suggests using number as length, thinking about numbers as lengths, which is a sort of way of making them concrete. 108 00:15:47,33.10892948 --> 00:15:50,752.10892948 And, again, it's actually a way of thinking about number that you never have to let go of. 109 00:15:50,752.10892948 --> 00:15:55,402.10892948 It's an entirely consistent way of thinking about number right until higher levels of mathematics. 110 00:15:55,512.10892948 --> 00:16:12,742.10792948 And again, within the NCETM materials I think one really fantastic innovation is that alongside introducing numbers as objects there is a strand through the professional development materials of dealing with number as length and so number as object number as length run alongside. 111 00:16:12,772.10692948 --> 00:16:21,352.10792948 And I think that there's a real power there in terms of offering students different ways into number and making links and connections across these two representations. 112 00:16:21,718.67459615 --> 00:16:30,198.77459615 So numbers as length, you might be thinking of something like Cuisenaire rods or bar model type representations. 113 00:16:30,318.77459615 --> 00:16:30,508.77459615 Yeah. 114 00:16:30,538.77459615 --> 00:16:31,408.77359615 It blew my mind... 115 00:16:31,458.77459615 --> 00:16:33,188.77359615 I once solved a problem. 116 00:16:33,578.77459615 --> 00:16:36,228.77359615 It was a GCSE problem and I was just playing around. 117 00:16:36,228.77359615 --> 00:16:37,118.77359615 It was on a podcast. 118 00:16:37,613.77459615 --> 00:16:39,63.77459615 And I solved it with a bar model. 119 00:16:39,503.77459615 --> 00:16:42,323.7745961 And then the hosts were talking about how they solved it. 120 00:16:42,323.7745961 --> 00:16:44,933.7745961 And one of them used a bar model and the other one used algebra. 121 00:16:45,183.7745961 --> 00:16:47,853.7745961 And I thought, bar model is algebra. 122 00:16:48,403.7735961 --> 00:17:02,253.7725961 How is that? I had no idea that that's actually what you're doing when you're, when you're solving some of these sort of, primary school problems with the bar model, it's actually algebra, which is just like, but there you go, complexity. 123 00:17:03,23.7735961 --> 00:17:15,83.7735961 I mean, in Gattegno's original curriculum he proposes you teach algebra before arithmetic, that you work on the sort of more abstract relationships between lengths and how lengths fit together. 124 00:17:15,173.7735961 --> 00:17:17,723.7735961 And only later do you then put, put numbers onto it. 125 00:17:18,603.7735961 --> 00:17:29,93.7735961 I remember when I was teaching in Year 6 and the children would know that they were going to be doing algebra and they'd say, when are we going to be doing algebra? Is it really hard? I can't wait to do algebra. 126 00:17:29,93.7735961 --> 00:17:39,772.8620315 And obviously they were terrified, I've just thought now that some of the problems we were already solving I would have loved to have been able to turn around to them and said, you did it last year in Year 5, you know, that was algebra. 127 00:17:41,92.8620315 --> 00:17:42,182.8620315 Missed opportunity. 128 00:17:44,772.8244139 --> 00:17:47,2.8244139 And that brings us to the end of part two. 129 00:17:47,652.8244139 --> 00:17:53,322.8244139 I hope our discussion of Alf and Nathalie's really fascinating and thought provoking book has left you wanting to read it. 130 00:17:53,852.8244139 --> 00:18:00,972.8244139 I think anyone interested in how children learn or how we might go about teaching mathematics successfully will find it such a fascinating read. 131 00:18:01,902.8244139 --> 00:18:09,602.8244139 Do come back and join us for the final third part of our conversation where we put your questions, shared with us on social media, to Alf and Nathalie. 132 00:18:10,142.8244139 --> 00:18:17,812.8244139 And in the meantime, we'd love it if you could share this episode with colleagues, like the podcast and subscribe to our channel wherever you get your podcasts. 133 00:18:18,572.8244139 --> 00:18:23,302.8244139 You can also hit the notification bell to make sure you are notified when the next episode drops. 134 00:18:23,772.8244139 --> 00:18:28,132.8244139 And finally, if you're on Instagram, do follow us at @themathspodcast. 135 00:18:28,652.8234139 --> 00:18:29,612.8244139 Thanks for listening.
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