Ems Lord, Director of the University of Cambridge-based NRICH project, explains how one primary school developed learners’ conjecturing and convincing skills through the challenge of solving live maths problems – and the motivation of seeing their solutions published on the NRICH website.  

What’s the problem?

Imagine the scene: your carefully planned problem-solving activity has completely engaged your class. They’re busily applying their mathematical skills in a real-life context and the higher attaining learners are being suitably stretched too. Towards the end of the session, you gather the class together to share their solutions. One of your high attainers raises a hand and suggests the correct answer. In response, you ask the inevitable question, “How do you know?” Instead of launching into a convincing argument, they simply shrug their shoulders and say “I just knew the answer!”

What are the three levels of conjecturing and convincing?

Developing the skills to conjecture and convince are essential components of our mathematics curriculum.  Nevertheless, even high attainers sometimes struggle to explain their thoughts to others. They might have convinced themselves about their solution, but they are not yet able to convince another person. They need time to explore others’ solutions and develop their own convincing answers too. In Thinking Mathematically (1982), John Mason talks about three levels of convincing: convincing yourself, convincing an enemy and convincing a sceptic.

Submitting solutions to "Live Problems" on the NRICH website

As learners make the journey from being a novice to an expert at mathematical reasoning, they will progress through several distinct stages. To begin with, their solutions might simply describe how they went about solving their problem. “We do train them that it’s either right or wrong, don’t we?’ noted a teacher.

Following a visit from the NRICH team, which focused on using solutions to develop reasoning skills, the school’s mathematics subject leader set every class the challenge of submitting their own solutions to a Live Problem. The teachers began the process by exploring examples of learners’ work already published on the website, ordering the solutions according to their level of reasoning and comparing their different content. This session was supported by the article The Journey from Novice to Expert.

Back in class, the teachers introduced their classes to the various Live Problems on the NRICH site, explaining that the learners could submit their own solutions. This was incredibly motivating for learners. “There was a real reason for doing it, a bit like when you’re writing in English and you want a real reason,” explained a teacher.  

Let’s look at one of the Live Problems explored by the learners. In Number Detective, learners need to identify a mystery number by following a list of clues:
  • The mystery number has two digits.
  • Both of its digits are even.
  • The digit in the tens place is greater that the digit in the ones place.
  • The ones digit is not in the three times table.
  • The tens digit is not double the ones digit.
  • The sum of the two digits is a multiple of five.

By focusing on explaining rather than describing their mathematical thinking, the learners developed their solution:
  • Amelia says, “8 and 120 are not the number because 8 is one digit and 120 has three digits.”
  • Aironas adds, “It can’t be 18 or 83 because they have odd digits.”
  • “46 and 22 don’t have a tens number greater than the units number, so it can’t be them,” suggests Matas.
  • Jessica states, “86 is not it because it has 6 in it (the 3x) and the rest are not.”
  • Tommy D says, “It isn’t 42 because the tens digit is double the ones digit.”
  • There are now only two possible answers left: 64 and 80.
  • Lastly, Tommy C goes for it! “I think 64 is the answer because 6 + 4 = 10 and 10 is in the 5 times table.”
As you can see, the learners carefully explained their thinking. “I thought it would be daunting for them to be able to justify why they’d chosen a certain answer, why they’d decided on a certain thing and made a statement, but I was really surprised by how many wanted to stand up and justify themselves,” noted one of their teachers. Seeing their solution on the NRICH website created a buzz around the school and beyond: “They couldn’t wait to actually go home and tell their parents all about it.”

How can this be developed further?

Whatever their current level of reasoning, learners can also try writing their own problems for others to solve. One very successful approach is using the NRICH problem as a template. Here are two new versions of Number Detective submitted by learners from the school, which have since been published by NRICH:

As the learners progress through their schooling, they will be able to start justifying their solutions by providing a correct logical argument that has a complete chain of reasoning to it. Their improved solutions will include words such as “because”, “therefore”, “and so”, “that leads to”...

Having seen some of their learners’ work published on NRICH, I asked the teachers if they encourage their learners to try more Live Problems . “Yes, I would love to, definitely. It was fun,” one of them told me. More Live Problems are uploaded every half term and learners can also challenge themselves with some unsolved Tough Nuts problems.

If you would like to develop the reasoning skills in your classroom, make sure your class know about the latest Live Problems, by subscribing to our free NRICH newsletter.

Become an NRICH ambassador…

NRICH is partnering with NACE to pilot its new ambassadors scheme through the NACE Research and Development (R&D) Hubs. Through this partnership, teachers at NACE member schools will have the opportunity to work alongside the NRICH team, exploring ways to share NRICH’s resources with schools at differing stages along the journey of developing confident and competent problem-solvers. Supported by bespoke training from NRICH, ambassadors will share their learning through the NACE R&D Hubs, collaborating to develop approaches for wider dissemination of NRICH’s resources for teachers and learners. 

Want to get involved? Contact NACE to find out more.

Ems Lord has been Director of NRICH since 2015, following a previous role leading one of the country's largest Mathematics Specialist Teacher Programmes. Ems has taught mathematics across the key stages, from early years to A-level Further Mathematics, and has worked in a variety of settings, including a hospital school. She’s supported schools as a leading mathematics teacher, local authority consultant and Chartered Mathematics Teacher, and has taught mathematics education on both BEd and PGCE teacher programmes. She is currently working on her PhD thesis, which explores approaches to improve support for those learning calculation skills, and is President-Elect of the Mathematical Association for 2019-2020.

Monday, February 19, 2018