Forward-Facing Primary Maths: What It Is And Why It Matters

Jo Austen

A call for primary teachers to make deliberate, future-focused choices in methods, models, and examples to support pupils’ long-term success in maths.

“What is 673 x 10?”

“Just add a zero!”

Most teachers know that this isn’t a great idea. As soon as we get decimals involved, this ‘quick trick’ becomes a real problem (673.4 x 10 does not equal 673.40). This is a simple example of ‘backwards facing’ maths, of an approach that may well accelerate students towards the current objective but is not ‘facing forwards’ to their mathematical future.

All maths teachers in all phases should aim to be forward-facing, looking ahead to the mathematics that students will encounter further down the road. I first came across the concept in Mark McCourt’s excellent book, Teaching for Mastery, where Mark reminds us that “an effective mathematics education system is one that focuses on teaching approaches that maximise subsequent progression, rather than a pursuit of short-term, superficial success.” What this means, beyond the general ‘prioritise learning over task completion’ point, is that we need to prioritise long-term learning, even if that makes things more challenging for ourselves and our students in the short-term. Essentially, we need to make choices that our students’ future teachers will thank us for!

For primary school teachers, I think this can be broken down to three areas in which to make deliberate, careful choices:

• Methods
• Models
• Examples

Methods

There are numerous ways of solving any given calculation. What’s important is that we don’t simply ‘settle’ for a child being able to get to the correct answer. For example, a Year 1 child calculating 8 + 3 by taking eight blocks, then another three blocks, then counting the whole lot to reach eleven, is an important step, but not something we can settle for. It is a slow, inefficient method and would, if relied on in future, become a problem, clogging up students’ limited working memory. The forward-facing teacher values mental maths, knows the progression of methods and is aware of the need to move children on to the more efficient ones.

Progression of methods for simple addition

• Count all
• Count on from the first number
• Count on from the greater number
• Derive from known bonds (e.g. 8 + 2 = 10 so 8 + 3 = 11)
• ‘Make 10’ method (7 + 5 = 7 + 3 + 2 = 10 + 2 = 12)

This is not to say that there is always one ‘best’ method to work towards mastering. Many primary children have the misguided idea that mastering a formal written method for the four operations is the ultimate aim. Far from it! We should explicitly teach children to choose an appropriate method for the question at hand, depending on the numbers involved. For example, 400,000 – 26 would be a very messy formal column method, but is straight forward if we have a sound understanding of place value. Similarly, doing 34 x 102 with a written method would be unnecessarily slow to those who recognise the simple mental route of (100 x 34) + ( 2 x 34) = 3468. Teachers need to explicitly model this kind of thinking, knowing that future, more challenging mathematical concepts will be far more accessible to students whose working memory is not consumed by calculation.

Models

The importance of physical manipulatives and pictorial models to develop deep conceptual understanding is now a widely accepted part of primary maths in UK schools. However, the forward-facing teacher goes further, understanding the progression of mathematical concepts well beyond the year group they are currently teaching and using this knowledge to inform their choices of models and manipulatives.

Example 1

A Year 1 teacher could comfortably rely on part-whole models to help their class solve simple missing number equations involving addition and subtraction. However, they might also introduce simple bar models, not because they are a necessity for the current objective, but because the teacher knows they will become extremely useful to pupils later on.

Example 2

A Year 4 teacher is tackling two-digit times two-digit multiplication. The objective could be met with a grid method and no use of an area model, but he does spend time exploring area models, first with dienes blocks, then with an easily adjustable virtual model, such as the one below from the superb MathsBot site. This forward-facing teacher knows how powerful the area model will be in secondary school when children come to explore quadratic equations.

Example 3

A Year 6 teacher is working on negative numbers with her class. To meet the Year 6 objectives, she could rely entirely on number lines. However, she instead also uses two-colour double-sided counters, introducing this model as she knows its utility for modelling ‘zero pairs’ in future. 

Examples

The examples we choose when exploring any given concept need to help us expose the full extent of that concept. If we don’t, we risk children developing misconceptions, storing up trouble for the future.

A good example of this is the need to present equations to children in different forms. It is extremely common to see something like 2 + 4 = 6 but do we see 6 = 4 + 2 as much? Subtractions in this form are even less common: how often do primary teachers expose children to 4 = 6 – 2? Failure to face-forward leaves secondary teachers to pick up the pieces, battling to reconfigure students’ mental models of concepts that have been built up over many years.

Another good example of this need to expose children to ‘unusual’ examples of a concept is in our teaching of shape. What makes a triangle a triangle? What makes a hexagon a hexagon? Using examples such as those in the image below will support fruitful discussion of key properties, embedding a much fuller understanding. The same could be done for something less visual like rounding numbers. What is 346 rounded to the nearest 20? Instead of following a fixed algorithm about which digits to look at and go up or down, exploring a question like that will help children to understand that the rounding question is actually asking, “which multiple of X is Y closest to?”

Conclusion

A great place to start with this is by chatting to teachers in other year groups. What does the concept you’re about to teach look like at their level? How might that affect the methods, models and examples you choose? However, the most important thing is the whole school culture around maths: forward facing primary maths is about seeing all teachers as responsible for all children’s mathematical futures, not just for getting them to the expected standard for the current year group. It is about teachers knowing the maths that children will tackle in future and making pedagogical choices to set them up for future success.