###### Math Fluency

With the rise in constructivist math instruction philosophy, we have seen a decline in popularity for teaching procedural and computational fluency within math instruction. With some educators going so far as to say that we should not teach basic math facts or procedural knowledge at all, oftentimes making statements like, “why would we teach students things that a calculator can do.” For example, long division is a procedural formula that can be quite difficult for students to use; however, students can use a calculator and instantly answer any division question. This of course begs the question does teaching students directly for the purposes of their math fluency impact their overall math achievement.

In 2019, Cason, Et al, conducted a meta-analysis on numerical competency instruction, of which procedural and computational fluency were a major focus. Their analysis examined 17 studies with 39 effect sizes. While their meta-study “prioritized” experimental and quasi-experimental design studies, they did not exclude studies without control groups. This could explain some of their outlier data and may have inflated the results. Their results can be seen in the below charts.

Definitions:

Number Sense: knowledge of the interconnectedness of number systems, procedural flexibility, accuracy, and competency.

Math Facts: fundamental math facts such as times-tables, addition, and subtraction sums, that allow students to answer questions with automaticity, without having to use procedures.

Numeracy: “ numeracy skills may include mathematical-logical thinking, relational reasoning, and specific concepts foundational for number sense such as one-to-one correspondence”

Discussion:

While, I do worry that this effect size might have been inflated, by not excluding case studies, overall this research does seem to suggest that teaching math fluency specifically is important. Only 4 of these studies showed below average results and only one study showed statistically insignificant results. Indeed the two lowest study results were also intervention studies, not classroom studies, which typically show lower results. In general we see that number sense is the most important form of numerical competence instruction, which makes sense as it includes the fundamental place value system, of which our mathematical systems are based on.

On the other hand, having students memorize math facts seemed to show the lowest impact; however, the impact of math facts was still not statistically insignificant. That being said, most of the math-fact studies were intervention studies, which typically show lower effect sizes. The only math-fact study included, that was a classroom study, was by Schutte, Et al, and showed a mean Effect size of .50. What this might suggest is that teaching math-facts in an intervention setting to students who may have working memory issues, might not be the best use of intervention time. However, I would assume, spending some time teaching math fact fluency, is an essential part of a balanced math program, especially for younger students.

Interestingly, teaching numerical competencies and math fluency appeared to have some of its best results in early primary, with diminishing returns as students get older. This makes sense, as numerical competencies and math fluency are foundational skills that students need to develop to fully understand our math system. However, this would be contrary to some postmodern education philosophy, which would stress more implicit instruction in younger grades and less implicit instruction in older grades, as numerical competencies are clearly best taught through explicit instruction. However, this does match our scientific understanding of language development. As research routinely shows that explicit foundational instruction is what’s most important for early language skills development.

Studies Included, as Summarized by Cason, Et al.

Anuio 2015:

“Development of numeracy (rational and counting) skills in low-, average-,

and high-performing Finnish kindergarten students Includes six effect sizes, two each for low-, average-, and high-performing students. Students took the Early Numeracy Test, and

each group of students includes one effect size for rational skills and

one for counting skills.”

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Burns 2012:

“Effects of computer-delivered math fact intervention on achievement of low-performing third and fourth graders in the U.S. Includes two effect sizes, one each for third and fourth graders. Each group of students took the Star Math assessment.”

Chard 2008:

“Effects of an experimental math curriculum on achievement of mixed-ability kindergarten students in the U.S. Includes one effect size for this group of students. Students took the Stanford Early School Achievement Test.”

Dyson 2013:

Effects of an 8-week number sense intervention on achievement of low-income, low-performing kindergarten students in the U.S. Includes two effect sizes for this group of students. Students took the Number Sense Brief and the Woodcock-Johnson assessments.

Ezbicki 2008:

Effects of a math fact fluency (addition and multiplication) intervention focusing on derived strategies versus traditional algorithms on achievement of mixed-ability fourth-grade students in the U.S. Includes three effect sizes for this group of students. Students took the Monitoring Basic Skills Progress assessment, which included one effect size for Computation Fluency and one for Concepts and Application, and they took the Group Mathematics Assessment and Diagnostic Evaluation assessment for Operations and Computation.

Graham 2013:

Effects of the QuickSmart numeracy program on achievement of indigenous and non-indigenous middle school students in Australia and development in average groups of comparison students to determine whether the intervention for low-performing students is closing the performance gap. Intervention & Non-intervention Includes seven effect sizes for multiple groups of students. The first four effect sizes represent low- and average-performing indigenous and non-indigenous students, the first two representing the intervention group and the second two the non-intervention group, who took the Northern Territory–developed assessments. The last three effect sizes represent low- and average-performing indigenous and non-indigenous students, the first two representing the intervention group and the third a combined non-intervention group, who took the Progressive Achievement Test in Mathematics.

Jordan 2013:

Development over multiple years using number sense as a mediator for mathematics achievement from mixed-ability first- through third grade students in the U.S. Includes two effect sizes for this group of students. Students took the Number Sense Brief and Woodcock-Johnson assessments

Hassinger: 2014

Development of low-performing kindergarteners in the U.S. to predict achievement in the first grade. Includes one effect size for this group of students. Students took the Number Sense Brief assessment

Lavelle 2013: Effects of a 15-week home numeracy program in two urban Catholic schools on achievement of mixed-ability first-grade students in the U.S. Includes one effect size for this group of students. Students took the Group Mathematics Assessment and Diagnostic Evaluation assessment.

Salaschek 2014:

Development over one school year analyzing the role of number sense in math growth trajectories for first-grade German students. Includes one effect size for this group of students. Students took a researcher-made assessment.

Schacter 2016:

Effects of a 6-week Math Shelf tablet intervention versus the most downloaded and best reviewed pre-K apps on achievement of pre-K students in the U.S.

Schutte 2015:

Effects of distributed practice (2× and 4× per day versus massed format once per day) of math facts on fluency of third-grade students in the U.S. Includes two effect sizes for multiple groups of students. The first effect size represents students who had distributed practice 2 times per day versus massed format once per day used as the control group. The second effect size represents students who had distributed practice 4 times per day versus massed format once per day used as the control group.

Somerville 2015:

Effects of an educational psychologist developed numeracy intervention in the UK on achievement (reasoning and operation) of kindergarten students. Includes two effect sizes for this group of students. Students took the Mann-Whitney U-Test, with one effect size each for reasoning and operation.

Sood 2011:

Effects of a number sense program in high-poverty schools on the achievement of kindergarten students in the U.S. Includes one effect size for this group of students. Students took the Early Numeracy—Curriculum Based Measure.

Toll 2012:

Effects of a numeracy intervention program in the Netherlands on the achievement of low-performing kindergarten students compared with the development of typically achieving students. Intervention & Non-intervention 0.57 (98) Includes two effect sizes for low- (intervention) and average performing (non-intervention) students. Students took the Early Numeracy Test.

Toll 2014:

Effects of remedial numeracy support to compare interventions of different lengths (complete and short) on achievement of low performing kindergarten students in the Netherlands compared with the development of typically achieving students Intervention & Includes three effect sizes for multiple groups of students. The first two effect sizes represent low-performing students who had complete and short interventions. The third effect size represents average-performing student.

Van 2015:

Effects of remedial numeracy education in the Netherlands on the

achievement of low-performing language-deficient and -proficient

kindergarten students. Includes two effect sizes, one each for language-deficient and

language-proficient students. Students took the Early Numeracy Test.

Written by Nathaniel Hansford

Last Edited, 2022-03-09

References:

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