In the midst of discussions surrounding our 'refreshed' curriculum and concerns about losing the diverse and open curriculum we once cherished, I believe it's important to approach new knowledge with an open and questioning mind. Embracing insights from global educational research does not necessitate abandoning the effective practices already in place within our schools. Instead, we can integrate new understandings to benefit our students' learning experiences, staying true to our local context while remaining open to the evolution of our practice.
Reflecting on my teaching journey, I've realised that a static pedagogical approach is counterproductive and static is certainly something I have not been over the last 27 years of teaching!
Education is not about swinging aimlessly on a pendulum but about continuously striving to improve as a practitioner. Clinging stubbornly to outdated methodologies, despite overwhelming evidence of better alternatives, is a disservice to both ourselves and our learners.
A New Perspective on Mathematical Fluency
Over recent years, I've embarked on a journey to understand the role of fluency in mathematics, particularly concerning basic fact recall. As a teacher of Year 1 and 2 students, my focus is primarily on addition and subtraction, although some students have begun exploring multiplication and division.
Before developing my scope and sequence in 2021, my approach was heavily influenced by the 'discovery' camp in mathematics, I believed that a focus on basic facts and any sense of timing not only caused anxiety, but completely ignored the 'beauty' of mathematics.
This discovery/inquiry method celebrated student-developed strategies and often delayed explicit teaching. My understandings were informed by the New Zealand Numeracy Project and the work of Jo Boaler, I have long since realised they limitations of both. My journey into Structured Literacy prompted me to reassess my mathematical thinking, acknowledging that while many practices had really merit, some of my previous practices were based on misconceptions.
Debunking Myths Around Mathematics Learning
Let's address some common myths about mathematics learning that I once held true to:
Myth: Timed tests cause math anxiety. Fact: Math anxiety is dispelled by ensuring students have the skills and foundations to perform the calculations.
Myth: Mathematics needs to be fun and engaging, and discovery alone achieves this. Fact: Success in mathematics is the most engaging and motivating factor. True enjoyment comes from the feeling of capability.
Myth: Problem-solving must precede foundational learning for authentic understanding. Fact: While problem-solving is crucial, foundational skills provide the necessary tools for effective problem-solving later allowing learners to generalise their understandings to new but related concepts.
Myth: Memorising facts is not genuine learning. Fact: Understanding relationships between numbers and having a strong number sense enables students to recall facts efficiently, to 'remember' them quickly, thus reducing cognitive load.
Myth: Focusing on basic facts discourages students from enjoying maths. Fact: My students' enthusiasm for mathematics has never been higher, with basic facts being just one component of our learning.
Discovering the Instructional Hierarchy
A pivotal moment in my journey was discovering the Instructional Hierarchy through the Knowledge For Teachers Podcast and the Chalk and Talk podcast. This framework, discussed by Sarah Powell and Amanda VanDerHeyden, clarified the science of learning for me. It outlines a structured approach to learning, starting with acquiring understanding through concrete materials and strategies, followed by building fluency, and finally, enabling generalisation and adaptation. I will not go into great detail here as a quick google will lead you to ample information written by people much more knowledgeable than I.
Building Fluency Through Structured Practice
My approach to building fluency first involves my scope and sequence, first written in 2021 and incorporating blocked and spaced practice.
This year, I've also focused on a retrieval practice schedule, significantly improving fluency in areas where students have acquired foundational understandings. I found that last year, I was not deliberate enough with retrieval practice, having a schedule has helped me keep on track with this.
Inspired by Brian Poncy, I have used taped tests and various retrieval practices to strengthen fluency, focusing not only on basic facts but also on numeral writing.
The Power of Relationships in Mathematics
Over the past two years, I've learned that fact learning without a solid schema is ineffective. Instead, students thrive when they understand the relationships between numbers. This understanding enables quick recall and builds speed through repeated practice. For instance, knowing that 2 + 2 = 4 helps students deduce that 2 + 3 = 5 and 3 + 2 = 5, facilitating rapid recall. When understanding relationships between numbers, children first need a strong sense of the 'value' each number symbolises, then using visual tools such as hundreds boards and number paths helps children to really cement these relationships.
Generalisation in Action
This year, I've witnessed the power of generalisation. By focusing on fluency in addition, students naturally generalised subtraction. Although we didn't explicitly practice subtraction retrieval, their understanding of the relationship between addition and subtraction allowed them to extend their fluency. Understanding generalisation has also helped me work smarter when it comes to fluency building, spending time on what will give learners the biggest bang for their buck, rather than going over facts that I know they will achieve fluency with without as much time spent on them.
This generalisation of subtraction evident in recent assessments where students demonstrated strong generalisation skills. In these assessments children had two minutes (three and a half seconds per fact.) Once the two minutes was up, the picked up a felt and completed the rest, this allowed me to assess fluency and accuracy.
Focus on Doubles and Bonds to Ten
Our journey began with an emphasis on doubles and bonds to ten. These foundational skills are essential for building mathematical fluency. Interestingly, we observed that facts such as +0, +1, and bonds with five, while requiring some specific focus, tend to develop naturally as children begin to understand the relationships between numbers. Therefore most of our retrieval practice focused on doubles and bonds to ten. Interestingly I have found that using a hundreds board has really helped children cement their doubles, knowing the counting in twos (in even numbers) pattern and attaching this to what we knew about the sum of a double, really helped for this understanding to be cemented.
The Role of Woodin Patterns
Woodin patterns play a crucial role in helping students quickly grasp facts involving numbers up to ten. By regularly engaging with these patterns, students are able to build quantities to ten effortlessly. This regular practice lays the groundwork for fluency in other areas. These are of course the foundation of my scope and sequence.
Expanding to Halves and Teen Facts
Once students master doubles, introducing halves becomes a natural next step. With just a bit of teaching, students readily grasp the concept of halves and needed to take up none of our assigned retrieval practice time. They already had the fact, they just needed to turn it around. Similarly, once the place value of teen numbers is introduced, children swiftly develop recall for these facts. Understanding the commutative property further enhances their ability to achieve fluency in a wide range of mathematical facts. Explicitly showing children that if they know 4+ 3 = 7, the know 3 + 4 = 7, frees up a lot of the cognitive load and means we don't need to practice both of these as seperate facts.
Assessing Fluency and Confidence
Written Tests and One-on-One Interviews
In addition to written basic fact tests, I conduct one-on-one interviews to assess students' declarative speed. Ideally, I aim for students to recall facts within two seconds, while also checking on accuracy beyond this time frame.
Impressive Results
The results have been impressive. Most students in the class are either fluent or accurate in all targeted facts, and they exhibit remarkable confidence. Even if some students do not achieve recall within two seconds, the majority manage to do so within four seconds, which is outstanding.
Below are some examples of these from a cross-section of the class, including a couple of Year One and Year Two students. The progress we've made underscores the effectiveness of our approach to building fluency in basic facts.
The Power of Fluency in Problem Solving
Fluency building has had profound benefits, especially in problem-solving. Last term, I noticed a significant improvement in my students' independence during problem-solving sessions, particularly within the change-based schema—an area that has traditionally been challenging. This newfound independence clearly aligns with their progress in mastering basic facts.
Rethinking Fluency: Beyond Memorisation
Building fluency isn't about rote memorisation. It's not about drilling facts into students' minds only for them to forget later. Instead, it's about employing the instructional hierarchy working first on acquisition. This means ensuring children take the time to master concepts and knowing when and how to build fluency. It's a gradual process, requiring spacing and numerous repetitions for many children to build speed over time. It is then about using quick assessments to establish when a fact is 'known' rather than blindly going over the same ones time and time again, once known, there is not a lot to be gained by repeatedly practicing this skill, instead our time is best spent on those facts that are taking longer to embed.
The Importance of Mastery and Generalisation
Fluency is about remembering and allowing for an economy of thought. It frees up cognitive energy to tackle more complex problems and enables students to generalise their knowledge across different contexts. With fluency, problem-solving and rich discovery approaches become more meaningful, as students have the foundational knowledge readily available to maximise these experiences.
I used to worry about a focus on basic facts causing anxiety for my learners, but instead the reverse is true, they are more independent and confident problem-solvers, and they engage more deeply with rich discovery-based learning because of their growing fluency in basic facts.
Instead of steering away from fluency building because of a fear we will cause anxiety, we need to realise that an anxiety towards maths is caused by the feeling of not being capable of doing the maths. There is nothing more motivating or engaging as the feeling of success.
