Scaffolding maths at Noarlunga

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One important principle underpinning this Make it Count project has been that we won’t just adopt the AL teaching sequence which was developed for teaching narrative. We would look at the theorists whose ideas inform AL, and bring those theories to this project. So now we’ll have a brief reminder of some of the theorists who are travelling with us, riding on our shoulders, whispering in our ears.

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Most important is our friend Lev Vygotsky who gave us the learning theory that underpins Accelerated Literacy pedagogy. Unlike his contemporary Piaget, Vygotsky understood that learning occurs in a social context. Not a party, but where culturally important events are happening. The process moves from intermental social engagement, with language being the most important cultural tool available to us, to intramental, when that language has been internalised. That learning then becomes a resource for further learning. We know that children are involved in many learning events to support them in moving to the next stage of development and that their success in moving to that next stage of development depends on us, as teachers, making the learning clear to them, and checking for handover. We also know that real learning occurs in the Zone of next development, with the support of culturally informed others. That’s where the pointy end of our teaching / learning negotiation happens. If children can do something independently that’s great, we’ve reached our goal, but we are no longer in the zone of proximal development.

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Then we have James Wertsch, who has done a lot of thinking about how we interact with the cultural tools available to us in order to be successful. A pole-vaulter who doesn’t know how to use the pole properlly is never going to break a record, no matter how good the pole is. A good pole-vaulter is going to be better, but they will be more successful with a carbonfibre pole than a stick of bamboo. The tools we have to use matter. A good pole-vaulter has had many years of training, learning everything they can from other pole-vaulters’ techniques, and perhaps then developing new techniques. As maths teachers, our students need to know what the goal is, what we want them to learn and why. We have to give them the useful tools, namely language and mathematical understanding, to help them achieve those goals. We are building common knowledge about mathematics in the classroom, so that we all become resources or tools for each other.

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Just as in teaching narratives in English our goal is access to literate discourse, in mathematics our goal is access to mathematical discourse. Our students have to think like mathematicians, care about what mathematicians care about, talk like mathematicians, joke like mathematicians. That’s the green circle. Now here are teachers in the middle. We have a bit of an idea, at least we know how to teach addition, but we’re not really in the discourse ourselves. So there’s the first challenge: we know a lot more now about how fiction writers think and we can talk about that when teaching narrative. What would we say about how mathematicians think? And here are the students. They are even further away from mathematical discourse than we are. How do we move us all along towards the world of maths? The challenge for us is to build meaning, and to do that we have to know what we’re talking about, ie we have to understand the maths really well ourselves, and secondly, we have to know how to scaffold students to help them think like mathematicians.

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Jerome Bruner and David Wood were also followers of Vygotsky. From them came the notion of scaffolding, where the adult does what the child cannot, always expecting, and checking for handover of understanding so that the scaffolding can be removed. Teaching and learning cannot be separated. They are two sides of the coin: the teacher’s teaching becomes the child’s learning. We call it ‘teaching /learning negotiation’ because that’s what we are doing all the time. We initiate learning, and through our conversations with the students, we work out what bits they get and what they don’t, and negotiate our way to successful learning. It’s also called the ‘pedagogic dance’: sometimes the teacher is leading, sometimes the child, until eventually the child can take over the leading. This is handover.

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Bernstein, a sociologist, analysed all sorts of discourses to look at their knowledge structure. Now we are all good at teaching literature. We understand how authors work. Although there are boundaries to that discourse, the knowledge and concepts are more or less horizontal; it doesn’t matter whether we teach suspense or descriptions first, or whether there are two or three characters in a book, or whether you work on poetry or narratives. There is lots of flexibility. This is where maths and science are very different. They are vertical discourses: one concept builds on another. You have to be able to describe a concept before you explain it. It matters where you start, and Rosie and Marie will be able to talk about the false starts we made last year because we didn’t really get this. Secondly, discourses have boundaries. We know when we’re in maths. The boundary is marked by mathematical language and mathematical materials. It is important that students know when they’re crossing from the common sense into the mathematical. It helps them understand that this is new learning, and it helps them understand what cultural tools to call on.

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New to our understanding is the world of neuroscience. Daniel Wllingham has written a very good book with a bad title ‘why Don’t Children Like School’. Amongst other important points, he explains why discovery learning does not work for so many students: Students are novices, they are not expert. So while you can send experts off into groups to solve problems and come up with novel ideas, it doesn’t work the same for novices. Experts have already acquired the cognitive and linguistic tools to solve problems. Novices don’t have these tools. That is what we have to teach them, and sending them off to discover for themselves is like sending an apprentice carpenter to build a house without a hammer and nails and tape measure. Do you remember in the original training we talk about ‘pointing students’ brains in the right direction’ as the purpose of our preformulating questions? Willingham supports that principle: we have to draw students’ attention to what matters. We have to show them what to attend to, and then we help them to build the meaning of the process or thing to which they are attending.

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And then we have Brian Gray, the developer of the Accelerated Literacy Program. Brian began his research by looking at the teaching of science texts with fringe dwelling Aboriginal students. His research moved to narrative because the discourse and language of science was so far removed from students’ every day lives that it was very difficult to build the bridge between them. The same could be said of mathematics: mathematical thinking is different from common sense and it is a real challenge to make the links. The scaffolding routine that you are familiar with in Accelerated Literacy is a routine developed to suit the teaching of narrative. Already, in a year of this project, Rosie, Carlie, Anna and Marie have refined and adjusted that routine to suit the teaching of mathematics. Although the principles are the same, the routine is not exactly the same. The use of preformulations, ie pointing student brains to what you want to attend to, is just as important in mathematics as it is in english. So are reconceptualisations: developing common knowledge about what matters. After the question has been answered, you are going to tell students why this is important. The challenge in mathematics is for us to know as much about maths, about why decisions are made, about the role of the mathematician, as we know about the authors of literature.

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From a Vygotskian perspective, language and thinking are very closely related. Students begin to use language even before they have a deep understanding of what it means, and that language becomes the tool for strengthening thinking. Students need mathematical language to help them solve problems. Just playing with concrete materials is not going to work. The process of becoming embedded in mathematical discourse is that the concrete materials are removed, and students just work with numbers on a page. They need mathematical language to do that. Here is the challenge for moving into scaffolding maths. Before you start, you need have identified what you want to come out of students’ mouths to show that there is handover. Not just a vague idea, but a really clear understanding. You should be able to write it down. That text becomes your guide for how you talk in your teaching / learning negotiation. If your students can’t explain their mathematical processes, then you haven’t focused enough on your target text.

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So here are some of the principles that have been developed from the 2010 iteration of the project. The teaching routine has been adjusted (I assume someone will talk about that separately) Included in the routine is talk about the ‘role’ of the mathematician: what do they think is important. This has been hard for us, because we didn’t have any idea beforehand, and it is hard for mathematicians to articulate this themselves. We begin where necessary with concrete materials, but if we are to move into mathematical discourse, they are a short-term prop. Mathematicians use language and paper. They work in the abstract, and that is what we are aiming to do. If we move too quickly to the abstract, then students don’t develop the concepts, so it’s a hard call to decide when to do this. We don’t stay with concrete materials though just because it is thought to be good pedagogy. Concrete objects are a tool.

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There are three important areas of teacher knowledge which are required for successful teaching of mathematics. The first is that you need to know your mathematics, and I know you are working on that this afternoon. A sub-set of that is that you need to know how to express this mathematical knowledge in valued ways. If you are going to move students into maths, you need to move from the common sense to the mathematical, and that includes language. How do you produce a high quality explanation in mathematics? What do mathematicians do when they define a maths concept? This is learning that will continue throughout the year. Finally, the scaffolding principles and strategies that you already know about from AL are important here too. As you work out how to preformulate in mathematics, you will know when you have been successful when you can say ‘yes’ to a student’s answer and than reconceptualise successfully. That will be one of the big bits of learning for this year. This project has been really exciting for all of us: some hard learning from things that didn’t work so well to start with, and lots of really positive learning as we worked out principles that seem to work consistently.