In this blog post, mathematics consultant Belle Cottingham argues that a maths mastery approach holds the key to ensuring more able learners develop the creative problem-solving skills needed for success – not only in exams, but in their future careers.

“We don’t do differentiation now, we do mastery instead!” I was recently told by a school teacher.

The more I discuss mastery with teachers, the more I encounter a general misunderstanding in the application of it. It seems to be a commonly held belief that when using a mastery approach, all learners must be doing the same work and move through the programme at the same pace. It therefore follows that those considered “more able” suffer from a lack of differentiation.

Is that what mastery is about? Shouldn’t the more able be accelerated? What should learners do in a mixed-ability class if they have finished the question? How can a teacher manage learners’ work when some complete the task quickly and others need more time to grasp the concepts?    

These are challenging questions for most teachers and educators, and at present there is little clear guidance.

Depth, not speed

Of course, the answer is simple: yes, there should be differentiation. But how can this be achieved in a mixed-ability class where teachers are often already maxed out in terms of time and resource?

Prior to the mastery approach, more able learners were usually accelerated by moving through the curriculum faster than others. For example, if most of the class was learning how to multiply, accelerated learners who knew how to multiply were taught how to divide.

Is learning a tick-box exercise? Does teaching someone to drive make them immediately a driver? Is the breadth and depth of a question important too?

Learning takes time.

When mastery is truly applied as it was intended, learners get the opportunity to explore and deepen their understanding beyond the boundaries set within the national curriculum.

There is always more to learn; there are always new ways to look at a concept. Learning is not a linear concept; it’s a curve that moves up and down, constantly changing. A process of evolution. Learning is not about memorising steps, or reeling out answers parrot fashion. It’s about developing ideas, making connections, clearly understanding the why behind the question, rather than racing straight to the answer.

Why we need maths mastery

In 2017 the pass boundary in GCSE mathematics (grade 4) for our learners was 17%.

17%! That’s what young people – our future nurses, dentists, architects, plumbers – needed to pass the exam. There was still a drop in the number of students that passed the exam. To achieve the maximum grade possible (grade 9), our most able learners had to achieve only 80%.

I wonder how we compare with our neighbours? Or other developed countries? Sometimes I think it is best not to know the answer, and this may be one of those times. In my view, 17% to pass and 80% to achieve the maximum grade is simply not good enough. Were the exams that difficult? Was the mathematics used so advanced? What is happening to our brightest learners from primary schools when they enter secondary school?

I recently worked with a group of 20 students in Year 11. Their GCSE grade targets were 7 and 8. I gave them the second question from a GCSE specimen paper. The first few questions are usually the ones everyone can access, but that’s not what I found at all.

Out of 20 learners in the group only three managed to solve the question independently. The other 17 said they were unsure what “proportion” was. They were considered “good” students in school (based on previous results). They wanted to do well in their GCSEs. They wanted to solve the question. They just didn’t know how. So where are we failing?

Developing creative problem-solving skills

Why are our more able learners unable to solve a question unless we ask them specifically what we want and tell them how we want it? What happened to initiative, taking risks, trying things, not being afraid?

There is no point asking learners to think critically, extend their knowledge and problem solve in exams if we haven’t invested time and energy teaching them how to do this. We can’t expect our learners to solve a question in different ways if teachers have been trained to explain a concept in only a single way.

Change doesn’t happen overnight. For teachers to embed new practices they need to see evidence that they work, and they need to know that they will have the time to implement change and see it through without external meddling. They need to buy into the idea.

This summer I was lucky enough to observe seven primary and secondary schools in Japan. They were all mixed classes of 30-40 students. All the lessons I saw were based around a single task. If the learner solved the task in one way, s/he immediately moved to a second method and then a third and so on. I was surprised to see that the more able learners didn’t look bored when they had solved the task in one way. They knew exactly what was expected of them – they just kept going, trying, drawing graphs, using algebra, exploring...

If this was possible in a classroom of 40 learners, it’s certainly possible in our schools. Frankly, Japanese results speak for themselves. Japan is consistently ranked in the top five countries for mathematics, according to TIMSS results.

What is particularly impressive when you analyse the TIMMS results is Japanese learners’ aptitude for solving questions they have never seen before. As can be seen from the illustration below, though only 54% of the material from the test had been taught in Japanese schools (different countries have different curricula), learners in Japan achieved an average of 69%. They could solve questions they hadn’t been taught because of their mathematical thinking/reasoning ability.

Is this an important skill in life? Well, a problem wouldn’t be a problem if we knew the answer beforehand!

Instead of accelerating children through the curriculum, the maths mastery approach supports deeper understanding and exploration – allowing more able learners to develop creative problem-solving skills that will serve them well not only in exams, but in their future careers.

Continuous improvement

All these ideas may be simple to apply, but knowing when to use them and how to use them can be challenging. Our teachers need the right resources, support and training to be able to adapt and grow themselves. It is simply not good enough to roll out a concept and expect people to organically learn. For our learners to solve tasks in multiple ways, they must be taught by teachers who themselves are capable of using different approaches to solve tasks. This may sound obvious, but a lot of teachers will have grown up in an education system where learning through repetition was rife. Consequently, structures must exist to provide guidance where this is needed.

For the maths mastery approach to be successful in the UK for children of all ages and abilities, we need to allow time for that to happen. Countries like Singapore and Japan have been working throughout the 1980s and 90s to shift their national approach gradually to a mastery model. What we see now is the result of years of relentless work from teachers, authors, parents and learners. Importantly, this process remains ongoing in these countries with frequent reflection and improvements/adjustments being made – which is of course consistent with mastery. There is always room to improve and learn more!
Mathematics consultant Belle Cottingham has a Masters in Mathematics and Learning, and a decade’s experience in teaching and tutoring maths for all ages and abilities. A member of the Mathematical Association, Association of Teachers of Mathematics, National Association of Mathematics Advisers, and Japan’s Project Impuls, she writes for the Mathematical Pie Magazine and has authored teaching guides and textbooks, including contributions to the Rising Stars Mathematics range. You can follow her on WordPress and Twitter.

For Belle’s advice on how to challenge more able learners within a maths mastery approach, click here.

New maths mastery resources coming soon…

For 2017-18, NACE is developing new resources to support primary maths teaching and learning for the more able. As part of this project, NACE is partnering with Rising Stars and NACE member schools on a research project to identify and share effective practice in this area. The results will be shared in the summer term, when NACE will also run a workshop on “strengthening talk in mathematics”.

For updates on this project, log in to the members’ area of the NACE website.

Not yet a member? Join NACE today.
Monday, November 27, 2017