Fractions Instruction

I have long thought teaching fractions is the most difficult strand of math to teach in the elementary grades. For starters fractions are conceptually challenging for students to comprehend. For example, many teachers can successfully teach their students how to multiply a fraction, but how many students really know what that means, or what it means to divide a fraction. This is problematic, because we want students to be able to take the procedures they learn in math class and apply them to real life problems. They cannot do this if they do not understand the concepts behind the procedures. Which brings me to my next point, the procedures for fractions are also complex and challenging for students. This is made even more difficult, once students realize that each type of operation for fractions is usually solved in a different way. 

 

A 2017 meta-analysis, by Ennis, looked at 15 experimental/quasi experimental intervention studies on fractions instruction focusing on students with exceptionalities. Students ranged from grade 3 to grade 9. However, most studies were in grades 4-8. While this study focused on students with exceptionalities, most of the studies included were specifically on ADHD students. Students with ADHD often struggle with learning procedural knowledge, because they struggle to focus long enough to retain all the steps in the correct order. Typically intervention studies show deflated effect sizes, however, this meta-analysis found multiple large effect sizes. While the study was not focusing on classroom instruction, it stems to reason that the strategies that benefited the students here would also benefit students in classroom interventions. I have graphed the results below, to better visualize the information. 

A note on studies included:

The original authors included in their result, a study on video modeling with an effect size of over 11. I excluded this study from my graph, because it’s obviously too large to not be considered an outlier. I think it is important to note that the author also included several studies by Fuchs, Et al, who I have noticed in the past often uses non-standardized assessments, which mark students on a rubric. This may have inflated the effect size on explicit instruction, as most of the studies here were done by him. That being said, all of the explicit instruction studies were high yield, the lowest effect size was found in a Fuchs study and the highest, by another author. I have included the effect sizes by each individual author, graphed below, so people can see how the meta-analysis effect sizes were found.

Term Definitions: (As Defined by the Original Authors)

Anchored Instruction: 

“Anchored instruction for fractions involves the framing of mathematical problems within relevant and practical contexts to help facilitate problem solving and computation skills (Cognition and Technology Group at Vanderbilt, 1997). This instructional approach involves teachers presenting students with videos, vignettes, or building projects; teaching students how to identify relevant information; and solving problems using fractions computation”

 

Explicit Instruction:

“Explicit instruction involves providing clear explanations, modeling steps or procedures, opportunities for supported and independent practice, ongoing feedback, and summative assessment”

 

Graduated Instruction: 

“Another approach to mathematics instruction that has been applied to fraction skills is the use of graduated instruction, or graduated sequencing, which entails presenting concrete, representational, and abstract (CRA)”. 

Strategy Instruction:

“Strategy instruction is another approach used to teach fractions that has demonstrated efficacy for students with learning disabilities and other populations (Graham & Harris, 2003). A strategy is a goal-directed process for completing a task (Keene & Zimmerman, 2007). Researchers examining the effects of teaching strategies for fraction skills have taken varied approaches, including the use of mnemonics (Test & Ellis, 2005) and cue cards (Joseph & Hunter, 2001). Zhang, Stecker, Huckabee, and Miller (2016) used a variety of strategies to help support students, including cross-multiplication, number line, and visual representation. In her previous review of the fraction literature, Misquitta (2011) found positive results across studies using strategy instruction.”

 

Discussion:

Ultimately we see strong evidence for graduated, strategy, and explicit instruction for teaching fractions. With the highest impact being for graduated instruction. In a graduated model teachers first present fractions manipulatives to students, then visual diagrams, and lastly introduce procedural knowledge. Of course graduated instruction is less of a strategy, and more of a lesson format (sometimes referred to as CRA). Similarly there is some crossover with strategy instruction as it includes visual representation. Ergo, if a teacher is using a graduated approach, they are still using strategy instruction. Explicit instruction is also a requirement for graduated instruction as it is required in the procedural stage of the graduated model. 

 

The meta-analysis found the lowest results for anchoring fraction problems with real world situations. However, the ultimate goal of math instruction should be for students to have the self-efficacy to apply the procedures they learn to the real world. That being said, it does not look like connecting the fractions problems to the real world, is a useful instructional tool in itself. 


It is interesting that this meta-analysis found such a high effect size for a manipulatives strategy, as the 2013 meta-analysis on manipulatives by  Carbonneau Et al, found some of its highest effect sizes for fractions instruction, while typically finding lower effect sizes in other areas. It seems likely that manipulatives are specifically helpful for teaching fractions.

It is also interesting that the authors looked at explicit instruction studies, but did not include any studies on inquiry based learning approaches, perhaps they could not find any. While the direct instruction effect sizes were high, ultimately it would have been nice to also consider the antagonistic strategy, as so many teachers advocate for inquiry based learning in math, for “deep learning”. 

Ultimately, this meta-analysis provides compelling evidence for a very specific model of fractions instruction. Based on this meta-analysis, I would suggest teachers start fractional instruction with manipulatives and visual representations to help build students' conceptual knowledge and explicitly teach the procedures secondly. While CRA is sometimes suggested as a lesson format, it does not have to be a lesson format, but can be a unit format. IE the first lessons of the unit are based on manipulatives and are conceptual. The middle lessons are done starting with conceptual review using diagrams and followed up with the explicit instruction of procedures. Lessons could also employ  a gradual release of responsibility, by including less and less diagrams, over time. I would suggest real world situational problems, are done in the final stages of a unit, once students conceptual and procedural knowledge reaches automaticity, so that they can further develop their application skills. 

 

Written by Nathaniel Hansford

Last Edited 2022-03-17

 

References

Ennis, R. P., & Losinski, M. (2019). Interventions to Improve Fraction Skills for Students With Disabilities: A Meta-Analysis. Exceptional Children, 85(3), 367–386. https://doi-org.ezproxy.lakeheadu.ca/10.1177/0014402918817504

 

 

Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380-400. doi:http://dx.doi.org/10.1037/a0031084