Why do students struggle with fractions? Are they inherently more challenging to work with than their decimal counterparts? Is there data to link students’ success with fractions to their overall success in math? Let’s dive into 3 reasons why students struggle with fractions, and what we can do about it as teachers.
What does the data say?
Multiple studies (this study published in Psychological Science and this Carnegie Mellon study) have found that success in Algebra and Calculus can be determined by how well students can work with fractions and division.
“The research team found that fifth graders’ understanding of fractions and division predicted high school students’ knowledge of algebra and overall math achievement, even after statistically controlling for parents’ education and income and for the children’s own age, gender, I.Q., reading comprehension, working memory, and knowledge of whole number addition, subtraction and multiplication.”
Robert Siegler
If you are a math teacher, these studies are your lived experience, so they aren’t actually that surprising.
So… why do students struggle with fractions?
1. Counterintuitive
Fractions are counterintuitive. Most of the rules and understandings about whole numbers do not apply to fractions.
- Whole numbers grow in value as the numbers increase. Fractions decrease in value as their denominators increase. Smaller denominators mean larger values.
- Multiplying whole numbers results in larger numbers. Multiplying fractions (usually) results in smaller values.
- Dividing whole numbers results in smaller numbers. Dividing fractions (usually) results in a larger quotient.
Students cannot answer whether their solutions are reasonable when they can’t really explain why fractions follow a different set of rules to begin with.
What do we do about it?
As middle school math teachers, I think it is easy for us to jump over fraction meaning because it was covered in elementary school so we can get right to our grade level standards. In the long run, you will find the time going over how fractions are related to whole numbers beneficial.
To do this, I strongly encourage using a number line (extend the number line past 1, so students can see how 9/8, 12/9, etc. relate to 1) to compare and order fractions. When I think of, compare, or do anything mentally with fractions, I still think in terms of their relationship to whole numbers. Example: 4 – 1⅞. To get from 1 ⅞ to 2, it will require a ⅛ jump. Then jump from 2 to 4 is 2, so my answer is 2 and ⅛.
2. Decimals are “Easier”
Why are decimals easier for students to understand? For one thing, decimals have common bases (10ths, 100ths, etc) while fractions only have common bases if their denominators are the same. In addition, decimals have the entire system of currency to support students’ understanding.
Are decimals actually always easier to work with? Absolutely not. There are times where working with fractions is easier (example: systems of equations, solving for x in an equation), but for comparing numbers and adding and subtracting, yes. (More examples come to mind – I would rather divide 2 by ⅛ than divide 2 by 0.125 anyday!)
What do we do about it?
Don’t force students to work with fractions over decimals because students can and should solve problems in a variety of ways. Ideally, students would pick one over the other because it makes more sense to do that in the context of the problem, not because they can’t use one proficiently.
How do we make fractions more friendly to students? Students need exposure. They have to fully understand what is happening and why. This means teachers must fully understand what is happening and the reason for it. In that same study –
“When asked to explain the meaning of fraction division problems, few US teachers can provide any explanation, whereas the large majority of Chinese teachers provide at least one good explanation.”
That was once me! Become an expert on fractions beyond the algorithms. Check out the links at the end of this blog post to make sure you can explain fraction operations in your sleep.
3. The Operations All Require Different Rules
When you multiply fractions, you can multiply across (⅔ * ½= 2/6), but you can’t do that with addition or subtraction (⅔ + ½ ≠ ⅗) because you need a common denominator. You multiply by the reciprocal when you divide fractions but only of the divisor. The standard algorithm for multiplication and division doesn’t explain much of the process and can be easily mixed up in students’ minds. If this is the only information you provide students (ei: rules), they will absolutely confuse them. I know this from experience.
When working with whole number operations, students start with models. This step in making something abstract concrete is called the CRA Framework (concrete, representational, abstract) and it is necessary in building conceptual knowledge and later fluency.
Here are some vetted Youtube Videos on how to teach all of the operations using models, and additional blog posts I have written on the topics. Our All Access subscription for 5th and 6th grade introduces all fraction concepts with models too!
- Adding and Subtracting Fractions with Models Video
- Adding and Subtracting Fractions with Models Blog Post
- Multiplying Fractions with Area Models Video
- Multiplying Fractions with Models Blog Post
- Dividing Fractions with Models Blog Post with freebie (that has videos)
When we make fractions visual and provide context for them in relation to whole numbers, students are successful. Do your tips struggle with fractions? What tips do you have for teaching fractions?
