Variance: adapting questions to retain mathematical value

The current maths national curriculum aims to ensure that all pupils can reason mathematically and solve problems. Despite these core objectives, the programmes of study are dominated by detailed lists of knowledge-based procedures that pupils should be able to do, such as draw and measure line segments or substitute numerical values into formulae. These lists can influence how maths is experienced in the classroom, leaving problem solving until last.

This means that only those who learn at a faster pace are afforded the luxury of applying their knowledge to valuable problem solving. The consequence of this is that problems are often designed to be very challenging and feel like only higher attainers would be comfortable attempting them. We argue that this does not have to be the case.

By identifying the characteristics of a problem, we can adjust the relative level of challenge and accessibility of the task, whilst ensuring that it still provides an experience that builds the same skills, confidence and strategies that made it worthwhile in its original format. Carefully designed variants of a task can be a mechanism that affords all students the opportunity to engage in problem solving and reason mathematically.

What this could look like in practice 

Consider this problem:

The diagram shows two rods, partly hidden by a metal sheet

Two sevenths of rod A is visible, and three tenths of rod B is visible.

State which rod is longer, justifying your answer.

The problem is opaque, providing a need to interpret the situation and determine what mathematical structures are relevant. In this case, the maths required at the core of the problem is whether 2/7 is greater or less than 3/10 of a whole, alongside a justification which could, for example, be verbal, numerical or pictorial. 

Let’s consider this variant:

The diagram shows two rods, partly hidden by a metal sheet

Three fifths of rod A is visible, and two thirds of rod B is visible.

State which rod is longer and use a bar model to support your answer.

This variant of the original problem alters the fractions provided, and provides a more scaffolded approach to justifying the answer. However, the opaqueness of what maths is required is still preserved, and whilst the justification is scaffolded, there is still a requirement to justify. The alteration of the fractions also lowers the access point once the maths is revealed. Comparing 2/3 and 3/5 is simpler, in part because the decimal equivalents of these fractions are more likely to be known, whereas the decimal equivalent of 2/7 in the original problem is not.

We have created a variant of a problem that makes it accessible to a wider array of pupils, whilst preserving features that keep it from being routine - it is still a problem rather than a procedure. We could of course adjust other elements of the problem too, to reduce or increase say, the opaqueness of the problem. In the variant diagram below, we maintain the challenge of working with 2/7 and 3/10, but by segmenting the bars we are guiding pupils towards what they could do to solve it - we are making the problem more transparent, but maintaining the other original characteristics. 

Furthermore, we can create variants that increase, rather than decrease the level of challenge around a particular characteristic. For example, we could increase the opaqueness of the problem further by requiring an even deeper understanding of structure: 

The diagram shows two rods, partly hidden by a metal sheet

2/7 of rod A is visible. If rod A is longer than rod B, what is the smallest value of x such that x/10 of rod B is visible. 

In this final variant, not only have we increased how opaque the problem is, we have also introduced an increased emphasis on working systematically to solve it by asking what the smallest value of x is. Across all of these variants, we are manipulating the characteristics of the problem to make it more or less accessible whilst maintaining core elements of what make it a problem. In doing so, we are providing ways to enable pupils across a wider spectrum of achievement to become meaningfully engaged in problem solving. In a curriculum where problem solving and reasoning are fully integrated, then variance must be central to considerations of task design. 

Please explore our resources to find further examples of problem variants that can be used in the classroom.