Blog: Dr. Helen Drury, Director of Mathematics Mastery

# How to differentiate tasks for depth of understanding – the meaning of the mean

Posted: 28/07/14

One of the questions we often hear in mathematics teaching is “How do I teach the same topic to a class of students with different levels of attainment?”Traditionally, differentiation means changing the content so that students with different prior attainment work on different tasks, possibly using different methods. In some cases, differentiation leads to accelerating high achievers through further topics, whilst offering more of the same to the middle or low attainers.

Instead of accelerating, we believe in developing a depth of understanding for pupils of all abilities, which requires teaching fewer topics but in much greater detail. This allows teachers to differentiate tasks with greater ease and success. Rather than accelerating high attaining students through topics, they are challenged to delve deeper into it and solve problems with very little structure. To support middle or lower attaining students, tasks can be scaffolded in different ways. This enables all students to access the same curriculum, working on the same concepts and skills at the same time.

The key to differentiating through depth of understanding is to plan carefully considered tasks and lessons. A good task should enable all students to access the content and to engage with it. It should be accessible to all students and open to a variety of problem solving approaches. A good task should encourage students to think deeply and to make connections to different areas of mathematics; it should inspire students to want to know more and to engage fully with the concepts being studied.

So how can we design learning tasks to meet the needs of all students?

In Year 7, we introduce students to the mean in the Autumn term. We do this as an application of division, enabling students to practice and apply their division skills in a new context.

Here are five tasks, with the same theme and content, but differentiated in different ways:

- Find the mean of the following set of numbers: 2, 3, 6, 7, 8, 9
- Create a data set with a mean of 6.
- These numbers have a mean of 6: 3, 4, 6, 6, 8, 9. Change two numbers so that mean stays the same.
- 2, 5, 5, 7, 9, and X have a mean of 6. Find X.
- X, Y and Z have a mean of 6. I now add the number 4 to the data set. What is the new mean?

These tasks increase in complexity in order to deepen all students’ understanding. They remove layers of structure and support as pupils move through the tasks and begin to offer open-ended problems around mean. In doing so, all students work with the same topic and are encouraged to extend their knowledge and understanding of the mean by developing a deeper appreciation of its properties and the underlying mathematics.

Differentiating for depth of understanding allows teachers the time to focus on a topic until all students have gained in competence. It allows students to access the entire curriculum and master the skills they meet, equipping them to face the challenges of future topics with greater confidence.