3 important steps to help students memorise addition facts
Teaching strategies for addition and subtraction facts is important, but strategies alone won’t help all students develop fluency. Instruction needs to provide a bridge from learning strategies to the accuracy required to engage in fluency-building activities.
In a previous post, I discussed why it’s important for children to memorise addition and subtraction facts. But achieving this is easier said than done. While lots of teachers understand the value of teaching mental strategies for addition and subtraction facts, many teachers find that some students still struggle to develop fluency even after receiving good-quality instruction in these strategies.
One of the key issues is that fluency-building activities require students to respond accurately to large sets of facts. We might teach the strategy of counting on from the larger addend, see students apply the strategy successfully and then think, Great, now students can focus on fluency. But from the perspective of some students, that is a large jump. If, for example, we define the set as facts where 1 or 2 is being added to a single-digit number, that is 32 different number combinations. From the perspective of an expert, this is no big deal - you just need to learn one strategy. From the perspective of the novice, that is a lot and they may feel under-prepared.
The solution? There needs to be an instructional phase that acts as a bridge between initial instruction in strategies and the accuracy required to engage in fluency practice. We can therefore think of instruction in addition and subtraction facts as involving three phases:
Strategy instruction - The goal is to develop understanding and reasoning proficiency; and to learn a mental strategy that can be applied to an identified set of facts. This is achieved through explicit teaching using concrete materials and scaffolds that are gradually faded out.
Accuracy building - The goal is to respond accurately and confidently to a set of facts. This is achieved through both ongoing practice in the strategy and incremental mastery in retrieving all facts in the set.
Fluency practice - The goal is to build retrieval strength (fluent responding) and storage strength (long-term learning). This is achieved through timed retrieval practice of the facts in a set and maths games.
Let’s have a look at these steps in more detail.
Step 1: Strategy instruction
Most teachers are familiar with strategies children can use to derive basic addition and subtraction facts. For addition, these strategies can include:
Counting on, e.g. 7 + 2 can be solved by counting 7… 8, 9
Using doubles, e.g. 6 + 7 = 6 + 6 + 1
Bridging through ten, e.g. 8 + 5 = 8 + 2 + 3
For subtraction, these strategies can include:
Counting back, e.g. 11 - 2 can be solved by counting 11… 10, 9
Using fact families, e.g. 9 - 5 can be solved by thinking 5 + _ = 9
Bridging through ten, e.g. 14 - 6 = 14 - 4 - 2
Subtracting from ten, e.g. 14 - 6 = 10 - 6 + 4
One pitfall in strategy instruction is neglecting to fade out the use of concrete materials and scaffolds. The concrete-pictorial-abstract model can be useful here to help teachers fade out the use of scaffolds. For example, the ‘count on’ strategy might be taught in the following sequence:
With counters
With counters on a number track
With a number track but no counters
With fingers but no number track
Without scaffolds
Step 2: Accuracy building
The goal of accuracy building is to respond accurately and confidently to a set of facts. If we try to progress students to fluency-building activities before this is achieved, students may struggle and their motivation may be damaged.
One way to build accuracy is to continue practising the strategy that was taught. Many would argue that enough practice in applying strategies eventually leads to fluency. And it is true, this is a common trajectory. However, most teachers would recognise that although this approach works for many students, other students continue to struggle.
One complication is that the relationship between understanding and fluency is not one-way. A recent paper by McNeil, Jordan and Ansari stressed the bidirectional relationship between fluency and meaning-making. This means that although developing understanding through strategy practice can support fluency development, it is also plausible that some students need to develop fluency in order to engage in deeper meaning-making such as the application of a strategy. Another study compared interventions focused on either developing strategies or memorisation and found similar effect sizes. One method was not superior to another. Taken altogether, the main implication is that practising strategies and fluent retrieval are both important, but how they support each other is complex.
One more complication is that we don’t really know the underlying mechanisms of fluent retrieval. When you recall the answer to 9 + 6 or 8 + 3, are you directly retrieving a fact from memory or are you very quickly applying a mental strategy? If we knew that fluent recall is actually lightning quick application of a mental strategy, instruction would focus only on strategy practice. If we knew that fluent recall is merely fact retrieval, we might focus on only drill and practice. The reality is that, for any person, competence in addition and subtraction facts is a combination of applying strategies and direct retrieval.
A common-sense approach is to provide ongoing opportunities for students to practice strategies and, at the same time, gradually develop students’ ability to retrieve all the facts in a set. This allows a typical developmental progression of strategy practice becoming fluent fact retrieval. But it also allows students who are less proficient at strategies to continue progressing in their learning of basic facts and perhaps being able to leverage that knowledge into developing deeper understanding of how numbers are related.
One pitfall of practising a set of facts is that too many facts are introduced at once. Weaker students become overwhelmed. For this reason, when implementing retrieval practice, ensure only a small set of facts are introduced at one time and that the new facts are practised alongside previously learned facts. One incredibly powerful technique that achieves this is called incremental rehearsal. In its traditional form, incremental rehearsal is a one-to-one flashcard-based intervention that introduces only 1 new fact at a time. In the whole-class incremental rehearsal intervention for times tables that I designed, 2 new facts are introduced at a time and it integrates strategy practice with direct retrieval practice.
Step 3: Fluency practice
Remember that prior to engaging in fluency practice, students should have developed accuracy and confidence in responding to a specified set of facts. Fluency practice involves any activities that require fluent retrieval of that set of facts. These activities can be in the form of timed activities and games (and even games that are timed activities). Some things to keep in mind:
Timed activities have an explicit goal of developing speed, which is a good thing.
Effective fluency activities have a high response rate (students are answering lots of problems in the given time) and feedback (students know when they are successful and receive help if they make an error).
Daily practice and record keeping help students to develop a growth mindset.
Varying activities so that different modes of response are used (verbal, typing, writing) ensures all students have access to fluency-building activities that work to their strengths.
Systematic instruction requires that fluency activities are matched to the sets of facts in which students have developed accuracy and confidence.
How can instruction in addition and subtraction facts go wrong?
The reason why instruction in addition and subtraction facts goes wrong is usually that one of the 3 steps outlined above is missing. Commonly, schools might teach strategies but don’t practise for fluency. As a result, students don’t develop confidence and their ability to progress in maths is impeded. Another common problem is that schools might engage students in games without ensuring students can accurately recall the requisite set of facts. This amounts to attempting fluency practice without accuracy and the result is wasted time and frustration for many students.
We can get basic addition and subtraction instruction right
There is still unhelpful debate about instruction in basic addition and subtraction facts amongst academics and education consultants pitting conceptual understanding against memorisation. There are still influential consultants downplaying the importance of fluency in favour of only talking about strategies. Only recently, there has been a well-publicised debate about whether it’s better to teach procedures or teach reasoning. Teachers in classrooms don’t have to take sides. We can teach so that conceptual understanding and memorisation support each other. We can teach strategies and build fluency. We can teach procedures and develop reasoning. We can do all the above through following three steps:
Strategy instruction
Accuracy building
Fluency practice
If you are looking for resources for that second step of accuracy building for times tables, I’ve shared a free resource for that here. In a future post, I’ll be sharing a similar resource for addition and subtraction facts.
If your school is looking to upskill in Retrieval and Fluency in Maths, you can book me to run a full-day workshop. More info here.
References
Nicole McNeil, Nancy Jordan, Alexandria Viegut & Daniel Ansari (2025). What the Science of Learning Teaches Us About Arithmetic Fluency, Psychological Science in the Public Interest.
Lynn Fuchs, Sarah Powell, Pamela Seethaler, Douglas Fuchs, Carol Hamlett, Paul Cirino & Jack Fletcher (2010). A Framework for Remediating Number Combination Deficits, Except Child.
