What elementary instruction misses about fractions
by Angela McIver, PhD
As I was meeting with my team this morning to refine our unit on fractions, we discussed how even the brightest students become confused about fractions over time.
Studies have shown that students make little progress in their understanding of fractions between 4th and 6th grade (when fractions are first introduced and then expected to be mastered). This may be the result of the unintended consequences of well-meaning instruction.
Below are 3 very common ways students develop misunderstanding of fractions because of the way fractions are taught in elementary schools.
Misconception #1
The larger the denominator, the smaller the fraction
This is probably the most common misconception students develop when it comes to fractions. Although the statement is partially true, it only relates to unit fractions (fractions where all the numerators are 1). We call this pizza/pie thinking because teachers often resort to a pizza or a pie to explain this concept, usually with accompanying drawings. I often hear teachers say “Think of a pizza. Would you rather have a piece of a pizza cut into four slices (¼ of a pizza) or 8 slices (⅛ of a pizza)?” Once students latch on to this way of thinking, they apply it to every type of fraction regardless of the numerator. This thinking can be demonstrated in this video of a high school student when asked to order four fractions based on their relative size. To counter this type of thinking, it is important for teachers to make explicit connections between comparing unit fractions with all other fractions.
Misconception #2
Fractions exist on the workbook page (but not in real life)
A fraction is a tool used to describe problems, situations and phenomena that exist in real life. But most students leave elementary school with a belief that a fraction is a problem that exists on the page of a workbook or computer screen. And, our inability to connect the learning of fractions to real world scenarios is hampering student understanding. I believe this is directly tied to many elementary teachers’ fear of math when veering too far off the page (or computer screen) can expose their own fragile understanding. In this article about student math understanding after the pandemic, a group of teachers learn of how one teacher with math anxiety often skips math concepts to leave for her students’ teachers the following year and others agree. If this happens two or three times in a student’s time in elementary school, it can have an enormous impact on their ability to transfer understanding from the page to “real life.” As math concepts become more complex in the upper elementary grades, teachers tend to stay close to the textbook, unintentionally denying students the opportunity to develop a deep and flexible understanding of fractions.
Misconception #3
A Fraction is a fraction, a decimal is a decimal and a whole number is a whole number
Because whole numbers, fractions and decimals (as well as percents) are often taught in separate units with separate practice problems, students can leave elementary school believing that these are completely separate concepts. Instead, students should build on the understanding that all whole numbers can be written as fractions or decimals (although not all decimals can be represented as fractions - these are irrational numbers). Care should be taken to make sure students develop the fluency to move between these different representations.
Understanding the ways that misunderstanding can develop in students’ thinking can help teachers become more confident and competent teachers in their classrooms. Supporting the deepening understanding of fractions among teachers may help address these gaps.
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