The Science of Maths Fluency

Maths Fluency’s Impact on Maths Anxiety

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Trying to problem solve in mathematics without basic fact fluency is like trying to read without being able to decode. Phonics isn’t the only thing students need to be able to read and comprehend text, just as fact fluency isn’t the only skill needed to solve complex math. But they both serve as foundational skills to achieve the desired outcome. Thanks to the science of reading movement, phonics is getting its well-deserved attention. Yet fact fluency is rarely discussed. Even more concerningly, it is sometimes seen as not important. But that's far from the truth.

Basic calculation fluency (accurate and rapid solution of single-digit addition problems and complementary subtractions) is the strongest correlate of success in mathematics in primary school and the most frequent symptom of difficulties in mathematics (Cohan, 2013).

The term ‘fluency’ in mathematics can often be carelessly referred to as old-school rote learning, or even pointless memorisation of facts. For the article, I will borrow a description of mathematical fluency from Toni Hatten-Robert’s (2023) research paper ‘The Need For Speed: Why Fluency Counts for Maths Learning’:

Mathematical fluency refers to the ability to perform mathematical calculations using well-rehearsed procedures quickly and accurately and includes the ability to recall facts to the point of automaticity. It also involves a strong understanding of mathematical vocabulary and symbols, as well as the ability to read and interpret mathematical expressions and equations. Fluency provides a foundation for higher-level mathematics skills needed for problem-solving, reasoning, and critical thinking, as well as real-world problem-solving while promoting efficiency and confidence. When students are fluent in basic mathematical skills, mathematics anxiety is reduced and a positive attitude towards mathematics is fostered.

To take it one step further, within mathematical fluency, there are two components; fact fluency and procedural fluency. Fact fluency is the accurate, proficient and automatic recall of maths facts. It is the computational skills of approximately 300 fundamental facts using the four operations. Procedural fluency is the accurate, proficient and automatic recall of procedures needed to solve maths problems. The standard vertical algorithm for addition is an obvious and sometimes controversial example, but maths is full of a range of other procedures students need to build automaticity in. 

Here’s the kicker… both fact fluency and procedural fluency are needed to develop a conceptual understanding and vice versa. There’s no one before the other. They’re all intertwined and should be taught at the same time, each of them strengthening the other.

Another term worth defining is automaticity. Stickney et al. (2012) say automaticity is the ability to deliver a correct answer immediately from memory without conscious thought, as opposed to relying on calculation. Without this ability to retrieve facts directly, students are likely to experience high cognitive loads and produce inaccurate work. As the complexity of maths progresses past single-digit multiplication, calculations like finding common denominators when adding and subtracting fractions become an extremely difficult task. (Baker et al., 2013)

Working memory capacity limits mathematical performance, but practice can overcome this limitation by achieving automaticity. (National Math Advisory Panel, 2008) Fluency with arithmetic operations is strongly linked to success in higher-level math, as it reduces the cognitive load during more advanced mathematical problem-solving activities. Proficient use of working memory is associated with improved academic performance across subjects, highlighting the importance of developing math fluency in educational curricula. To give perhaps an obvious example, imagine trying to solve 3-digit multiplication or division if you haven’t developed fluency in basic multiplication facts. Working memory would be overloaded with basic facts whilst trying to build conceptual and procedural understanding. Every high school maths teacher is well aware of this problem that is caused by a lack of fluency building in early education.

Automaticity and fluency are often used interchangeably by mathematics educators. However, it is worth distinguishing the difference. Automaticity is answering accurately and with speed, typically within 2-3 seconds by most conservative definitions. Some research would even suggest it’s more instantaneous. Fluency requires a deeper understanding. It's not necessarily effortless, but it is efficient, flexible and accurate. 

Now we need to get into some controversial territory. If we acknowledge how important maths fluency is, we need to be able to accurately assess and monitor students’ facts and procedural fluency. Luckily, there is a heavily researched and easy-to-use tool to do so; timed fluency assessments. Timed assessments serve as a means for students to demonstrate their ability to recall information rapidly and accurately, providing educators with insights into their level of mastery. It's frequently observed that students with strong reading skills tend to make fewer errors during timed assessments, a correlation also evident in research on timed math fluency. Monitoring fluency, which encompasses both accuracy and speed, requires the regular administration of timed tests to gauge progress towards automatic recall effectively.

Before you start pulling out the stopwatch and preaching to your educator peers, it's worth noting that timed assessments are an area of contention amongst educators. If you bring up timed maths assessments in the wrong staffroom you would be in for a heated discussion. You would quickly hear the comment ‘timed assessments cause maths anxiety’. But, research such as Grays et al. (2017) article for the Journal of Behavioural Education would dismiss such an argument. 

In the academic literature, there is discussion around the relationship between ‘maths anxiety’ and ‘maths performance’. The direction of this relationship is often researched with many different theories suggested. Firstly, there is ‘The Debilitating Anxiety Model’ which suggests that maths anxiety leads to students being unable to learn and recall maths facts. Which in turn, leads to poorer outcomes. Secondly, there is ‘The Deficit Model’ which suggests that people who start with poorer maths skills will develop maths anxiety over time after a lot of failure and difficulty in the subject. Finally, there is ‘The Reciprocal Theory’, which says maths anxiety and maths performance have a bidirectional relationship. Anxiety leads to poor performance and poor performance leads to anxiety. A vicious cycle. 

If you look at any meta-analysis on the subject, it will typically conclude the reciprocal theory is most accurate and that more research should be done. Gunderson et al., (2018) research says children who start school with lower levels of math achievement not only lack some of the foundational math concepts that set the stage for later maths development, but they are also more likely to develop maths anxiety and less adaptive motivational frameworks. In turn, those who have higher maths anxiety and more entity motivational frameworks are more likely to achieve less in math over time.

Beyond the research, common sense and personal anecdotes would also lead you to a reciprocal relationship. But even though a vicious cycle between maths anxiety and maths performance is somewhat obvious, let’s not forget how students enter school. They aren’t born with maths anxiety. Students don’t start school being anxious about maths. Gunderson et al.'s (2018) research made this fairly clear.

Let’s clarify exactly what causes maths anxiety, and it isn’t timed fluency assessments. What causes anxiety is a lack of skills, particularly a lack of basic maths facts at an early age. That lack of skills and anxiety will have a reciprocal relationship and will likely spiral. On the other hand, what we know about timed fluency practice and assessment is they are an effective intervention method and can improve maths performance (Fuchs et al., 2021). Better maths performance would mean less maths anxiety.

So contrary to popular belief, timed fluency assessments may reduce maths anxiety. Try bringing that up in the staffroom.

The discourse around mathematical fluency, particularly regarding fact fluency and procedural fluency, underscores their critical role as foundational skills in mathematics education. Contrary to misconceptions, fluency is not about mindless memorisation but rather about efficient, accurate, and automatic recall of mathematical facts and procedures. This fluency, encompassing both speed and understanding, is essential for fostering confidence, reducing anxiety, and enabling higher-level mathematical problem-solving and reasoning.

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References

• Baker, Austin T. and Cuevas, Josh (2018) "The Importance of Automaticity Development in Mathematics," Ge Researcher: Vol. 14 : Iss. 2, Article 2.

• Cowan, R. (2013). The development and importance of proficiency in basic calculation

• Gunderson, E. A., Park, D., Maloney, E. A.,Beilock, S. L. & Levine, S. C. (2018) Reciprocal relations among motivational frameworks, math anxiety, and math achievement in early elementary school. Journal of Cognition and Development

• Grays, S., Rhymer, K., & Swartzmiller, M. (2017). Moderating effects of mathematics anxiety

• Fuchs, L.S., Newman-Gonchar, R., Schumacher, R., Dougherty, B., Bucka, N., Karp, K.S., Woodward, J., Clarke, B., Jordan, N. C., Gersten, R., Jayanthi, M., Keating, B., and Morgan, S. (2021). Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades. , Institute of Education Sciences, U.S. Department of Education.

• Hatten-Roberts, T. (2023), The Need for Speed: why fluency counts for maths learning. The Centre for Independent Studies. Analysis Paper 57.

• VanDerHeyden, A. M., & Codding, R. S. (2020). Belief-Based versus Evidence-Based Math Assessment and Instruction. Communiqué (National Association of School Psychologists)