Back to the Basics: The Power of Procedural Fluency in Mathematics

One of the most important skills when it comes to learning mathematics is procedural fluency. Procedural fluency in mathematics is, simply put, the knowledge and ability to apply mathematical procedures with ease. Take addition for instance—if I asked what 2 + 2 equaled, you would almost immediately recognize the answer as 4. That’s an example of procedural fluency! You understood the procedure and were able to apply it effectively.

The Importance of Procedural Fluency

At its core, procedural fluency is a concept that extends beyond the realm of mathematics. Think about riding a bike, for instance. The first lessons would be something along the lines of mounting a bike, then using the brakes, then pedaling, and so on. This is a procedure that, only after extensive practice, one is able to perform “fluently.” This analogy reflects two important takeaways about math fluency:

1. Knowledge is NOT the same as application. If you were to memorize the precise instructions on how to ride a bike, could you do it as an inexperienced rider? Probably not. And even if you could, it wouldn’t be nearly as effective as someone with experience. This is a common pitfall in many classrooms, where a lack of complementary practice can diminish the efficacy of even the best lessons.
2. Practice is essential. Just like riding a bike, one of the best and only ways to master procedural fluency is practice!

Why Should We Care About Procedural Fluency?

Automating Simple Tasks Frees Up Cognitive Load

Cognitive Load Theory (CLT), developed by John Sweller in 1988, is a well-established concept in educational psychology. It emphasizes that our working memory has a limited capacity, and when this cognitive load becomes too heavy, students struggle to perform basic tasks or grasp complex concepts. One effective strategy for reducing cognitive load is the use of organized patterns of knowledge called schemas. In math, schemas represent familiar procedures like multiplication tables, rearranging fractions, and other operations developed through repeated practice and procedural learning.

Now, imagine tackling the word problem below without these automated schemas:

Sally received some apples. Another person doubled her apples, then someone gave her 6 more. She gave away half and was left with 6. How many apples did she start with?

While this problem isn’t exceptionally difficult, there are many steps to setting up the equation (2x + 6) / 2 = 6 and ultimately solving for x = 3. Even if a student understands the correct procedures they must apply, consciously working through each step compounds the chance of an error at any particular point. Moreover, the cognitive overload can make it difficult to maintain focus on the bigger picture. Students can get lost in the sub-steps and forget the original question, wasting time and sometimes failing to solve the problem altogether. By automating these fundamental skills, students free up their cognitive resources to understand and solve more complex problems efficiently.

Procedural Learning Can Support Conceptual Learning

Many schools often focus on building a strong foundation through conceptual learning, where understanding the "why" behind mathematical concepts comes first. Only after grasping these ideas do teachers usually believe students are ready to handle procedural tasks effectively. Take multiplication, for example. In Western curricula, students learn that 9 × 7 means adding 9 seven times before they start memorizing times tables. To a lot of students and teachers, this approach feels intuitive and ideal.

But here's the interesting part: research shows that procedural learning can actually enhance a student’s understanding of the subject. This is demonstrated through a classic study by Rittle-Johnson and Alibali in 1999. In this study, students were divided into three groups and were given a lesson on the procedure for solving mathematical equivalence problems like 12 + 3 + 2 = 12 + __, the concept of mathematical equivalence, or no lesson at all. While students taught conceptually were able to apply procedures better, as expected, those focused on procedural practice were surprisingly also better at explaining the concept.

This approach can be especially helpful for more conceptually complicated topics like factorization. Instead of pushing students to fully understand every detail of a concept upfront, starting with a preliminary explanation, moving to practice, and then going back to a conceptual explanation can be a useful strategy. This suggests that procedural learning through worksheets and other means can be effective and introduced at many different stages of learning.

Procedural Learning Builds Confidence

Many students often develop negative attitudes toward math and doubt their abilities. They begin to lack confidence and believe they are “not smart enough” to excel at higher levels of mathematics. This mindset contributes to a fixed belief that improvement in math is impossible—and it’s not entirely their fault. The Western education system places a strong emphasis on conceptual learning, often at the expense of procedural practice. While conceptual learning is valuable, it doesn’t always provide measurable milestones. This lack of clear progress markers can make it difficult for students to see how well they are grasping concepts or understand where they need to improve.

Procedural learning, on the other hand, provides a straightforward path to progress: practice. In the context of worksheets, speed and accuracy are two clear metrics to assess improvement. By tracking these and practicing consistently, students can see tangible progress, boosting their confidence and shifting their mindset from fixed to one of growth.

Conclusion

Overall, procedural fluency is an integral part of mathematical competence. It simplifies complex problems, supports conceptual understanding, and most importantly, helps build confidence! To improve your procedural fluency effectively and for free, check out numbric.com today!