In this second of two blog posts, mathematics consultant Belle Cottingham outlines five approaches to ensuring more able learners are effectively challenged within a maths mastery curriculum.

In my previous blog post, I argued that a maths mastery approach holds the key to ensuring our more able learners develop the creative problem-solving skills they need for success – not only in exams, but in life more generally.

Of course, this is all great in theory, but how do we effectively achieve differentiation in maths mastery? There isn’t a strict right or wrong answer, but here are five approaches to try…

1. Anticipate and adapt.

Good teachers anticipate. They know their learners and anticipate what they may say, what mistakes they may make, what answers they may give. All learners are different, with different strengths. Just because a learner is more capable at calculation, doesn’t necessarily mean that s/he is also more able at problem-solving or shapes.

Anticipating answers is not easy; it takes years of experience and constant growth and development from the teacher’s point of view. However, the more accurate the anticipation, the better the tasks and the more appropriate the challenges the teacher can set.

2. Use skilful questioning to promote conceptual understanding.

Mastery is not about doing repetitive questions. In fact, the beauty of mathematics itself, with or without mastery, is that it is infinitely stretchable. Questions can be solved in more than one way. Questions can be asked in more than one way.

For example, let’s imagine a group of children are learning the 8 times table. Some will be quicker than others. Some may already recall the tables. Just because they can recall them, however, doesn’t mean that they understand why.

“Why is 8 x 3 the same as 3 x 8?” “What does 8 x 6 look like?” “Is 8 x 6 > 6 x 9?” These are just some of the ways the question can be asked or extended.

Each of these questions will make learners think beyond the simple calculation. A calculator can calculate; a brain can reason, question, explore… Brains were built for exactly that!

3. Use problems that can be extended for more able learners.

The choice of tasks and questions used in the classroom should be carefully considered and selected. The questions should be set so everyone in the classroom can readily attempt them, falling within the overall knowledge bracket, but they should also be suitable for simple extension to challenge and deepen understanding.

Continuing the tables theme, a question like “Find different ways to calculate 12 x 4” can be very rich in answers.


 
Some students may add 12 + 12 + 12 + 12, making links between addition and multiplication.



Others use the multiplication facts that already know. The 2 and 10 times tables are taught before the 12 times table. Hence, they can calculate 2 x 4, 10 x 4, then add the results.


 
Or they can simply use the properties of multiplying by 4, double, then double again. 12 x 2 = 24 and 24 x 2 = 48.



There are many ways to think about multiplying two numbers, and each of them can link to other ideas, concepts and applications.

4. Use concrete pictorial and abstract (“CPA”) representation.                

More able learners can benefit as much as their peers from the use of CPA representation to visualise and represent mathematics in different ways.

Providing concrete material for everyone will facilitate more able learners’ need to meet problems which are presented in different ways, in different contexts and with use of more varied vocabulary. Using the table question, more able learners may use counters or marbles to explain to a partner what 6 x 8 looks like. Being able to articulate the mathematical thinking is a very important skill that we need our future mathematicians, engineers, teachers and doctors to have.

More able learners may also be encouraged to work in mixed-ability groups and asked to write a question based on a picture they see, or write a question that has a mistake in it… The options of extending a mathematical task are limitless and the more it happens, the more robust the mathematical foundation in our learners will be.

5. Allow time to explore, think and reflect.

This is very important for all learning to happen. The mastery approach provides this. Reflecting on mistakes that a learner has made herself, or that someone else in the classroom made, is a very good strategy that can be used to clear any misconceptions, and is particularly effective through the learners’ own voices. Having time for reflection is crucial in creating maps of knowledge that can be used in developing future concepts or embedding the roots of the existing ones.

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.


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.
 
Date: 
Tuesday, November 28, 2017