What we’ve got wrong about learning mathematics | Alf Coles | TEDxBath

For the past 27 years I have been mulling over the question of how

learning

mathematics

can become relevant to students' lives connected to things they are interested in and meaningful to the world in which we live. I don't think there's a single way this can still be done, but today I want to share with you some of

what

I've concluded so far, and as I thought about those questions, the questions became relevant to me 27 years ago, like newly qualified teacher, when I started at a school in south London, my first job. in england, at a comprehensive school in south london, as a

mathematics

teacher, i recently qualified with a degree in mathematics and philosophy from oxford cambridge teacher education.

I had ideals for my classroom. I wanted my students to be creative and independent in their work in mathematics. I imagined them asking questions about the topic, discovering about themselves, thinking about

what

place they could fit in the world and what they could do with math and other things they were

learning

. I imagine teaching as an engine of social change towards a more just, more equitable, more sustainable society. Perhaps inevitably, reality turned out a little different in that first year of teaching. I admit that I discouraged many mathematics students, the students who could have I started with enthusiasm and ended up with much less at the end of the year.

I wanted my students to be autonomous, but perhaps I hadn't taken into account that autonomy could be expressed by ignoring me. My own journey to becoming a teacher has been greatly influenced by engaging with the writings and ideas of an African-born educator named Caleb Goteno, who actually would have been 110 a week ago today if he were still alive. I want to show you a 45-second clip of him speaking towards the end of his life. I hope the sound works. It is often repeated that I teach them, but they. don't learn well if you know that stop teaching don't resign yourself to your child stop teaching the way he doesn't read reach out to people to learn and try to understand what needs to be done so that you become more skillful every day to help These young people They fill their minds with things that are so elementary that when today it takes them five years I can do it in 18 months less.

Sometimes they stop teaching in a way that doesn't make people learn well. I certainly needed to do something different afterwards. my first year of teaching and it was through collaborations particularly with Lorinda Brown and others that I began over a few years to get a sense of how I could be in a classroom so that my classroom began to approach some of my ideals. Here is an image of a couple of pages of some students' work from 1998, about four years after I started teaching, I placed an emphasis on students asking questions and writing down their ideas about their mathematics.

I helped design a curriculum at this school where students would work on fairly extensive tasks over a period of time, a typical task might be to work on your own question for half an hour and the next lesson would be to share what the students had discovered and discovered and we would move on from there. I wasn't teaching for five years and 18 months still working on it, but most of the students were pretty committed now they teach at a university trying to support other people, other teachers, to make their dreams for their classrooms come true, but the ideas Coutinho's practice shows us what could be possible in schools, what we need.

I understand that we have to become more skillful every day because that's really what I want to talk about today in research I've done with Professor Natalie Sinclair, who works in Canada. We have come to the conclusion that there is a set of beliefs that blind us to some of the learning and the ways in which learning occurs. around us and today I want to talk about just one of those beliefs. It's a belief I believe. can inhibit people from reaching their potential in entire areas of life, a belief that has become dogma because it is unquestionable and seems quite innocuous, it is actually the belief that learning is some kind of vertical construct and, particularly , in learning mathematics, what we learn is a kind of building block.

By the way, we need solid foundations, you need to really understand the basics before moving on to anything else, and since this idea seems so natural, it also seems natural to us that in schools we often separate children depending on where they are in this vertical. construction and teaching them in different classes teaching them different things and only letting a group of students have access to the most interesting levels of mathematics is because this idea seems so natural that it seems obvious to us that if a student has not understood step nine We cannot teach them step 10, 11, or 12.

And what that means is that when you get to high school, if you are underperforming in math compared to your peers, you will likely be offered an essentially circular curriculum. where you keep revisiting this site of failure, this step that you don't understand, you never have access to what might be happening elsewhere in the curriculum, so I want to try to unravel this idea today because I think we know if we look around and we know that learning is often not like that. I think looking around us many of you might have experienced a toddler taking their first steps, if you haven't there are some wonderful trick clips on YouTube where you see this happening and a common feature. of many of these clips is that the child stutters a little starts to fall and then runs only a couple of steps usually caught later by an adult so if we think about that what it means is that children, including you, ran before we walked and even go so far as to say that we learned to walk by running, to take an example from history: astronomer Caroline Herschel, born in 1750, discovered comets and the catalog of stars, famously never knew her multiplication tables and However, it seems that his math building was missing.

It is not irreparably damaged by this shaky foundation: school-age children who go to a school where they do not know the language achieve functional fluency in three months, and that is not because someone decides which parts of the language they have to learn first and which . It's not about deciding what is simple and what is complex and charting some journey of your learning, but about being immersed in this complex whole, this complex situation and being able to use what Caleb Geteno would call the powers of your minds. Katena believed that if you have learned. In a language, then you must have had the power to detect patterns, make predictions, decide how to concentrate on a situation, be able to concentrate on one aspect and not another in a complex situation.

So what I want to think about in the second half of What I'm going to offer today is what this might mean for teaching. How might we teach in a way that respects this idea of ​​the powers of children's minds? And the image of learning that I want to offer is one much more connected to the topic. of this day of interconnections because it seems to me in that image of learning a language in that form of immersion, what you are learning is not a vertical construction, it is much more, you learn small knots of effective practice, small knots of relevance and you begin. putting them together creating more of a mesh or a network than a vertical construction and it is this image that I want to think about how we could teach if we are interested in learning as a kind of mesh, not as a construction, well what I want to offer two concepts in the vocational curriculum in mathematics which are often difficult for people and the first is negative numbers, they cause a lot of difficulty for people so this chart is one that you will see in many primary schools in England. and it's very useful, we can use it to add numbers in units, add in terms, etc., but if you start and this is all you see about the number, eh, and if the number is tied to concrete objects when you start, it seems pretty It's hard to think of what a negative number could be.

This graph seems somewhat complete. It's not really clear where it goes from here. A very simple addition to me transforms what we're seeing there by adding these additional layers. There are questions that immediately arise, like what. What would happen if we keep going left or right, suddenly there are infinities there and I can guarantee you that if this graph was in the classrooms in England, even if you just focused on the white squares at some point, The child is going to tell you what are those numbers at the top with a little dash next to it and it seems to me that as soon as a child asks you a question, my job as a teacher is half done, is to look at a more complex situation.

That the idea of ​​negatives fits into some kind of pattern allows us, as students, to make use of that power of our mind to detect patterns or take another algebra example again, an obstacle in many people's learning of mathematics and usually at the beginning of algebra, you may be asked to think about what a plus is and another way of writing, what a plus b means, it all seems extraordinary, really arbitrary, what is going on here and whether students have difficulty with this type of mathematics , then again we could I often think well, maybe they need something simpler, maybe they need something concrete and if as a student you continue to struggle with this type of math, maybe you've actually decided that you've reached your mathematical ceiling, maybe you haven't anymore.

I'm going to try to teach you. These things maybe you're just not ready for it, you're not capable of it and I want to suggest that none of that is the case, that rather than needing something simpler, what might be needed is something more complex later on in the learning process. algebra, you may come across the idea of ​​algebra in the context of a function, so here we go. I'm going to do this now and imagine, well, we'll try to do this in a little bit of silence. You have to try. and find out what's going on here, okay, anyone figured it out, so I put that number in there.

If we were in a classroom, I would like someone to quietly come up and write it down, but I can't really do that. number that they think could get there, okay, if you said nine, you would be right and again, if we were in my classroom, I would ask you to give me the next number, it was that six, you said yes, great, okay , anyone wants to say what they think. this one okay and one more see if you can see what's going on here five okay so what's going on here? Even if you've never done algebra, I think I could put that one in and we could think about what's going on. the rule you're using here, how did you get that and if you haven't seen what the pattern was then maybe this will help you, so all the algebra is saying is that I'm taking my number, maybe you were adding it to itself and adding. one or maybe you were doubling it and adding one in this more complex way of thinking about algebra algebra as functions starts to make a little more sense is there some pattern to observe and notice this a more a and whatever it was suddenly became becomes a little more obvious, well yes, n plus n is just 2n, if I add a number to itself, it will be the same as doubling it.

Well, I want to offer you one more story. A few years ago I was supporting the master's dissertation of a teacher by teaching. in bristol and she was working with a group of underachieving 15 and 16 year olds, who were approaching their school final exam at 16. and decided she wanted to work with them on some ideas that were two or three years old. ahead of the curriculum in terms of any kind of sequential learning sequence and there was one particular girl in this class who had extremely low levels of numeracy and this teacher decided that she would work with them on the Pythagorean theorem that some of you you'll remember it was about squares and square roots and triangles, so this teacher worked with this class on the Pythagorean theorem for a couple of weeks about a month later, the teacher asked a question about the square root of 49 this girl raised her hand and answered He told her seven, the teacher almost fell off a chair because that was the first question this girl had answered in her lessons.

This god had probably been taught about square roots for the last seven years in his math teaching and probably what would have happened is that a teacher would have diagnosed that. she didn't know about square roots, she would have focused on it, she tried to fill this gap in her math learning base and it didn't work for seven years, two weeks working on something more complex and a month later she had retained what roots square roots it was by looking at this more complex situation that the square roots had a need, they had a purpose and became meaningful to her, something she could then retain,So what does it all mean?

What I want to suggest is that if you are a teacher and you are teaching someone who has found something difficult, instead of thinking that they need something simpler, it could be that they need something more complex and I guess that applies to you too. As a parent, if you are a student who had difficulty in math, please do not believe that there is any intellectual failure on your part. Most likely, you were never offered a picture of the complex whole into which the mathematics you were learning could fit. You were never offered the opportunity to use the powers of your mind to detect patterns, to make predictions about what was happening, and for all of you, the next time you think of learning as some kind of vertical construction process, I hope the image of a mesh functions as a network of disordered knots of relevance also comes to mind thanks