Conjecturing, convincing and getting their solutions published in 2018!
Ems Lord is the Director of NRICH and one of the authors of ‘Approaches to Learning and Teaching Mathematics’.
Take a moment to ask yourself, ‘Which of my sessions made the greatest impact?’ For me, it was a teacher workshop designed to encourage students to develop their convincing skills. Developing conjecturing and convincing skills is an essential aspect of active learning in mathematics but many students remain unconvinced by their own answers let alone be ready to convince others! Our project aimed to address those concerns with the ultimate aim of getting their solutions published. The outcomes of the project exceeded all of our expectations.
The school had already addressed its mathematics curriculum and devoted considerable time to professional development too. But, its students remained inactive learners. We aimed to redress the balance.
Our first step was rethinking what we meant by convincing. We looked at John Mason’s (1982) research on convincing and proof in mathematics. In the first instance, students need to convince themselves about their answers. Then, they need to convince a friend. The final test, the third level, requires convincing a sceptic. Too often, though, they were unconvinced of their own efforts and failed to reach even the first of the three levels. The students seemed a long way off convincing others.
At the teacher workshop we explored a range of rich mathematical tasks which would allow their students to work in different ways to achieve a solution, increasing the opportunities to discuss their ideas and evaluate their different approaches.
You can try similar activities with your classes, the free NRICH teacher resources for both Steel Cables and Kite in a Square are both examples of resources with multiple solutions. You can explore with your students, challenging them to present them to others and they’ll realise the importance of thoroughly understanding the solution themselves. At this point, they’ll working at the second level of convincing. But, our challenge was to reach the higher, more rigorous third level. The students need someone more challenging to convince.
At the start of every term, the NRICH team upload a new set of live problems for students. We also welcome their solutions and publish some of the best ones. The Live problems offered an ideal opportunity for the students in our project school to convince a sceptic and work at the third level of convincing. After all, if they could convince the highly regarded NRICH team to publish their solutions then they would be well on their way to becoming mathematicians themselves. So, we explored the latest set of Live Problems at the workshop and the teachers chose the most suitable activity to share with their students.
Over the next few weeks, our office started to receive submissions from the school. Nevertheless, we only publish solutions which explain the thinking behind the approach, we do not just look for the answers. The team shifted through the submissions and quickly realised just how much the students had learnt through the project. They were no longer just convincing themselves, many of them were able to convince others too. It was a really uplifting time, reading their solutions and considering which, if any, we might publish. NRICH has a huge international audience and we receive hundreds of solutions from around the world, choosing the best is never easy.
So, what happened next?
Despite the multitude of solutions swamping our office, we did not publish just one of their solutions, we published several. Each successful student had thought clearly about their reader, explaining their approach and often making excellent use of visuals. Their solutions are now live on NRICH.
What will be your major goal in 2018 for your classes? Could you make them more active too? Perhaps you could challenge them to try a Live Problem and share their solution. Maybe, just maybe, they will be the ones who get published in 2018.
Source: Cambridge International Examinations