Math Fact Fluency/Computational Skills

Math skills can likely be developed into four main types: conceptual, procedural, computational, and application. Conceptual knowledge is the understanding of the rules and on which math is based. Procedural knowledge is understanding how to use algorithms to solve math problems. Computational knowledge is the ability to solve simple arithmetic questions.  Application knowledge is the ability to take all of the other types of math knowledge and apply it to real world situations/new math problems. As an example, conceptual fraction knowledge tells us what a fraction is, and why we need to use different algorithms. Procedural knowledge tells us how to use algorithms like cross multiplication to solve fraction questions. Computational knowledge tells us how to do the arithmetic questions within a fractions question. Application knowledge tells us how to take all of that knowledge and apply it to a real situation. 

 

Research shows that conceptual knowledge and procedural knowledge are likely the most important, because they have the broadest base of applicability. However, conceptual knowledge has been shown to be the most important of the two. Likely because if students conceptual knowledge is strong enough they can either create their own math procedures or research a procedure to solve a math problem. Computational knowledge on the other hand is the most elementary of these skills. Moreover, computational knowledge can be substituted with a calculator. 

 

Computational skills, or math facts fluency, as it is sometimes referred to in the literature are often the subject of immense debate. During the late 90’s and early 2000’s we saw a rise in constructivist scholars who argued against the importance of computational instruction, in favor of more conceptual and application instruction, arguing that conceptual and application knowledge are more important. Ultimately the end goal for math instruction should be to empower students to be able to take the abstract math they learn in life and apply it to real world situations. However, in order for students to apply their math knowledge, they need to have developed the underlying skills which are conceptual, procedural and computational skills. 

 

Ultimately, serious math instruction scholars realize that this debate of which types of knowledge should be taught is a false dichotomy. All four types of math knowledge need to be taught. Moreover, while these skills are separate and do require some specific instruction, they are also inherently linked. Computational instruction helps develop conceptual, procedural, and application knowledge, and vice versa, as was shown by the 2019 Cason meta-analysis, and the Rittle-Johnson literature review. True math fluency is a result of the development of all four skills. The real question is not if we should teach each skill/knowledge type, it is how much of each should we teach, how best do we develop each skill type and when, should we focus on each skill type.

 

Scott Methe, Et al published a meta-analysis in 2012, on the subject of when and how best to teach computational knowledge. Their paper looked at 11 studies. While the inclusion criteria required a control group, many of the studies had very low sample sizes, so they used an IRD effect size calculation to compensate. That being said, I included this paper in my analysis, because it was one of the only meta-analyses I could find on the topic. However, the low sample sizes do diminish the reliability of the results, even with the use of corrective effect size calculations.

Definitions: 

Speed Based Interventions: Are skill and drill activities that encourage fast response times, such as timed work, math minutes, or Around the World. 

CRA: Refers to concrete-representational-abstract. This is a planning format that has teachers use manipulatives first, diagrams second, and skill and drill/procedural third. This method combines conceptual, procedural, and computational into one format. It is also an example of an iterative strategy.

Accuracy Focused: Math fact fluency exercises that promoted accuracy over speed.

 

Fluency Focused: Math fact fluency exercises that promoted speed over accuracy. 

 

Contingent Reinforcement: Reward systems for math success. 

 

Cover-Copy-Compare: Students fold a sheet of paper in three. They copy down ten math facts from the board in the left column. They fold and cover these math facts. They copy the facts again from memory. They then compare, to see if they get it right. 

 

Combined Approaches: Teachers used multiple of the other strategies listed, rather than focusing on one strategy. 

 

Interspersal: Teachers mixed the difficulty level of questions, providing some easy, moderate, and challenging questions. 

 

Discussion:

Overall this meta-analysis shows that teaching students math facts is a high yield strategy. Indeed every methodology had an average or above average effect size. The effect sizes for Speed based interventions, accuracy based interventions, and CRA were all especially high. It was interesting that interventions that focused on speed but not accuracy did less well. This probably shows that speed based interventions, which are not assessed in some way, are less valuable. It is also interesting that CRA did so well. CRA is not a fluency intervention in my opinion. It is a planning method that aims to teach three primary math skills at once. That being said, the CRA effect size here should be taken with a grain of salt, as it was based on a single, low sample size study. 

 

Similar to the Cason 2019 meta-analysis, this meta-analysis showed the highest results in early grades and the lowest results in older grades. This makes sense when we consider that computational knowledge is likely the most foundational knowledge for elementary school math. Teaching students how to add, subtract, multiply, and divide, gives them basic skills that can be applied to solving all other math. However, as students get older, and gain more mastery of this, it becomes less and less likely that further computational instruction is beneficial. The same trend can be observed in reading science. Where phonics is very conducive to learning in the primary years, but not after. 

 

To further examine this point I conducted a secondary analysis of the two meta-analyses on grade based outcomes. The results of which you can see below. 

As you can see from this chart, there is a clear and obvious trend. Teaching number facts and math fluency has a clear benefit all throughout elementary school. However, the returns of that benefit is clearly diminishing, as students get older. I think this suggests that the older students are, the less math fluency instruction they should be given. 

 

Written by Nathaniel Hansford

Last Edited 2022-03-25

References: 

 

Methe, S., Kilgus, S., Neiman, C., & Chris Riley-Tillman, T. (2012). Meta-Analysis of Interventions for Basic Mathematics Computation in Single-case Research. Journal of Behavioral Education, 21(3), 230–253. https://doi-org.ezproxy.lakeheadu.ca/10.1007/s10864-012-9161-1

 

Cason, M., Young, J., & Kuehnert, E. (2019). A meta-analysis of the effects of numerical competency development on achievement: Recommendations for mathematics educators. Investigations in Mathematics Learning, 11(2), 134–147. https://doi.org/10.1080/19477503.2018.1425591