Written by Steve Thornton

I want to expand a bit on the third key element of the reSolve Protocol: **reSolve classrooms have a knowledge-building culture**.

I want to make a distinction between the phrase knowledge-building culture and learning. It is possible to learn something without it necessarily being of much value, although I did once win a prize at a quiz night by knowing that the first name of Colonel Sanders of KFC fame was Harland. I might call this kind of knowledge ‘unproductive knowledge’, as there is little or no opportunity to produce new knowledge from it. I am sure we can all think of examples of mathematical knowledge that remains unproductive for some of our students.

On the other hand, a knowledge-building classroom develops ‘productive knowledge’, through which we can make connections, solve problems, make sense of the world and generate new ideas. In his book ‘Education and mind in the knowledge age’, Carl Bereiter^{1} writes that the distinguishing features of a knowledge-building culture are the belief that all knowledge can be improved, and the commitment to do so.

Bereiter identifies six characteristics of a knowledge-building culture.

- There is a focus on ‘conceptual artefacts’; that is, the concepts and ideas of mathematics itself, rather than on correct answers, methods or even solving problems.
- These concepts should strike us as interesting, and raise questions and have implications beyond those that are immediately apparent.
- The goal is for shared understanding rather than individual learning.
- There is a shared commitment to expanding our collective knowledge.
- Criticism should be both offered and welcomed as a way of refining and advancing knowledge.
- Being open to new ideas and beliefs, even if they take us outside our comfort zone.

To illustrate how these ideas play out in one of the reSolve resources, I would like to talk about one of our new resources. ‘Lunch Lap’ arose from a workshop in Toowoomba, focusing on Pythagoras and trigonometry.

Lunch Lap starts by describing a situation in which a sausage cart and ice-cream cart are placed on the two short sides of a 160 m by 120 m rectangle, and a drink cart on one of the long sides. There is a gate 40m from the corner of the remaining long side. Students investigate where to put the three carts in order to minimise the length of the ‘lunch lap’.

Five aspects of the knowledge-building culture that we have tried to capture in the resource stand out.

- There is a shared commitment to improvement. Students initially choose where they wish to place the carts, calculate the length of the lunch lap and compare. Gradually they develop a sense that the shape of the shortest lap seems to be a parallelogram.
- Students offer criticism to advance knowledge. They look at each other’s suggestions and try to improve them.
- We promote openness to new ideas. Students come to see that Pythagoras’ theorem doesn’t actually solve the problem, as the best we can hope for is a trial-and-error solution.
- New ideas are introduced by looking at the problem from a different point of view. With careful scaffolding, students see that by reflecting the rectangle several times they can create a shortest path by drawing a straight line between the gate and its reflection.
- The observation that the shortest path seems to be a parallelogram can now be explained and justified so that students develop a shared understanding.

Through the activity, students’ understanding is refined and improved, leading to active knowledge that makes connections between different areas of mathematics, and results in a mathematically convincing and elegant solution. Of course, we might argue that this is a very contrived and artificial situation, but then the power of mathematics, and of active knowledge generally, is that it can be used across contexts to solve problems that we may not yet know about.

^{1} Bereiter, C. (2005). Education and mind in the knowledge age. Routledge.