Procedural Fluency and Conceptual Understanding

In this series of posts, we will explore the history, research and practice of two key strands of mathematical learning: procedural fluency and conceptual understanding (sometimes referred to as procedural knowledge and conceptual knowledge, respectively).

What is the best way to teach maths?

Education has long been embroiled in a fierce dispute around how students best learn mathematics. It is a debate that consumes educators and policy makers the world over and has persisted for several generations.

In its most extreme form, the so-called Math Wars of the past fifty years (a laughable term if not for the very real threats made by some parties) has divided education stakeholders in the US. At its core is a fundamental misunderstanding of how to nurture mathematical thinking and an outright rejection of findings from education research.

Flick through a series of policy documents on mathematics education and you will likely notice an emphasis on two terms: procedural fluency and conceptual understanding. You’ve just met two unwitting parties to the most absurd debate in all of mathematics education.

procedural fluency and conceptual understanding boxing gloves

In the red corner…

Name: Procedural fluency
What it means: “skill in carrying out procedures flexibly, accurately, efficiently, and appropriately” 1
View from the corner: Procedural fluency is championed by many as the core essence of mathematical learning. In their view, it is critical that students acquire basic skills and develop fluency in order to engage with the world around them. The historical persistence of skills-based curricula is some measure of how persuasive this view has been in policy circles.

In the blue corner…

Name: Conceptual understanding
What it means: “comprehension of mathematical concepts, operations, and relations” 1
View from the corner: Proponents argue that deep thinking is only possible by first understanding of the structures that govern mathematics, and acquiring knowledge that is rich in relationships between concepts. The extreme view claims that procedural fluency is a distraction from this goal of higher-level learning.

A false dichotomy?

Both views have their merits and each is supported by several decades of research. The futility in the debate lies in the assumption – assumed too often by both sides – that these two strands of learning are somehow in competition with each other, and that focus on one strand jeopardises development in the other. In fact, history teaches us the complete opposite.

In the US, the New Math movement of the 1960s (seen by many as a response to the rapid scientific progress made by the Soviet regime) focused almost exclusively on conceptual understanding; demanding that students were taught mathematical concepts from the ground up, justifying every assumption with clearly stated logic. Needless to say, this attempt to impose a more rigorous curriculum was so extreme that it brought only confusion to parents, teachers and, ultimately, to students.

At the other extreme, advocates of procedural fluency have vehemently opposed any emphasis on conceptual understanding. A striking example is the antireform movement that rose against the effort to bring more problem solving into US curriculum standards back in the 1980s. A slew of stakeholders, including textbook providers, policymakers and even impassioned parents, aggressively pushed for a ‘back-to-basics’ approach that focused on skills and procedures. And so it goes that the past two decades have been dominated by a focus on procedurally-oriented learning tasks.

PISA UK rankings for mathematics

The result? Sliding standards that have left both UK and US students lagging behind their international peers. Both countries now fall outside of the top 25 for mathematics in PISA’s international ranking of education systems – the image to the right demonstrates the UK’s marginal performance in mathematics, according to the 2012 PISA results.

If history teaches us one thing, it is to avoid extremes.

Where are we today?

The latest round of curriculum reforms is wrapped in the language of balance between procedural fluency and conceptual understanding. From the Common Core State Standards:

Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

Meanwhile, the stated aims of the 2014 National Curriculum for England include:

The national curriculum for mathematics aims to ensure that all pupils ‘…Become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.’

In principle at least, there appears to be alignment between these frameworks and the longstanding wisdom of education research.

So is this a turning point for mathematics education or is history repeating itself once more? Before we can answer that, we have to first dive into the education research to better inform our understanding of how students learn mathematics.

The next post in this series will look at what the research tells us about fluency and understanding, how they depend on each other, and why they are both vital to developing a complete learning profile.


1 Kilpatrick, J., Swafford, J., Findell, B., & National Academy of Sciences – National Research Council, W. n. (2001). Adding It Up: Helping Children Learn Mathematics.


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