Construct

Richard Noss

Watching young children make sense of the world teaches us an important lesson: that people learn best when they are making things, and sharing what they’ve made with each other. Making something produces something to talk about, reflect upon, and ultimately learn with. And it presupposes that one has something with which to build – blocks, or paints, or musical instruments.

In the past, most things were more or less unbuildable – playing around with the forces of gravity, or modelling the spread of diseases with pencil and paper has always been out of reach for the vast majority. But now, with computers, literally anything is possible.

Computers open doors which used to be closed to everyone except the very few – the mathematicians and scientists and musicians who could build things in their heads. Now anything is potentially buildable on a computer, and if it’s buildable, it becomes thinkable, discussable, and ultimately, learnable. As Seymour Papert said, in describing his theory of ‘constructionism’ some 20 years ago(Papert, S (1980) Mindstorms: Children, Computers, and Powerful Ideas. (Basic Books: New York)), the special thing about building is that it constructs a ‘public entity, whether it’s a sand castle on the beach or a theory of the universe’.

In similar vein, Douglas Thomas and John Seely Brown in 2009 put forward a new model of education that fuses learning as reflecting, learning as making, and learning as becoming. Creative play and improvisation are essential for prospering in a complex and changing world.

The programming systems in Chapter 11 give some potent examples of what can happen if people – even very young people – are given powerful tools that allow them to construct and share ideas embodied in things (including virtual things).

Reflecting on what you have constructed is a key part of learning. Until now, this lesson didn’t easily translate into learning more generally. But now, with computers, ideas that could only live in the minds of people can have a life on the screen – bringing them alive, and, most importantly, giving people the chance to construct mental representations of dynamic systems alongside virtual ones.

 

 

Mathematics is the science of patterns. Identifying, analysing, and predicting patterns is the source of the power of mathematics – whether it’s a sequence of numbers, the structure of shapes, the change in the climate, the spread of a virus. But finding patterns in a few cases is not enough for mathematicians: the trick is to express the pattern so that it’s true for all cases – to generalise it.

This turns out to be difficult for learners. The problem is that if you ask someone to spot a pattern, say ‘how many tiles do you need to make a “train track pattern”?’ a natural strategy is to count. Why not? We can encourage people to think a little more by asking them to predict the number of tiles when it’s very large, but even then, asking ‘how many?’ cues people into counting in some way.

The trick is to look at the structure of the pattern, to see it as something that is repeated to form a rule. The classic way to do this is to use algebra – call the number of ‘building blocks’ N and go from there. But that is precisely what we are trying to teach! We’re asking learners to use the language of algebra, before they understand what that language is supposed to be about.

MiGen is an intelligent, computer-based support system intended to help pupils get to grips with algebra without the difficulty of manipulating abstract symbols. Young people enter a ‘microworld’ that encourages them to construct patterns in a colourful, dynamic and visual format. They are nudged to explore the nature of relationships and uncover rules for themselves. And they are empowered to present their answers creatively, using simple sequences of coloured tiles.

In the MiGen microworld, students bump into powerful ideas. They learn to move from the specific to the general. The important lesson here is that students first construct patterns, and only then, when they have built a computer ‘model’ of the tiling pattern, do they have to express what they have constructed in some form. That form is a sort of algebra, but it looks a bit different. It is learnable because it gives students a language to talk to each other about what they have made. In other words, the system helps students to see the general in the particular, to keep hold of the link between what they’ve constructed, and the rule that expresses that construction. After three or four lessons using MiGen, studies in five schools showed that students were able to apply their knowledge to conventional generalisation tasks.

MiGen adds a new dimension to the idea of construction, through its artificial intelligence techniques that support teachers. Intelligent systems that focus only on the students can end up marginalising teachers. But MiGen keeps them in the picture by providing a suite of tools to monitor students’ progress and to view and compare their constructions. Two students may, for example, see a pattern in two different ways, both of which are correct. They may both arrive at an algebraic expression for their pattern. But unless they know what the other has done they will not realise that rules that look completely different can in some sense be the same. MiGen can spot the potential benefit of their collaboration and help teachers group them together.

As well as highlighting fruitful collaborations, the system can help teachers gauge students’ progress, and pinpoint those in need of assistance. So the idea of constructionism is extended to harness the techniques of AI, to help students construct what matters, to notice what goes wrong, and – most critically – to reflect on and share as productively as possible, what they have built.

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