Students in the U.S. have an enormous deficit in their understanding of fractions. For instance, half of the eighth graders in this country are not able to correctly order 3 fractions, even though this is a 4th grade standard. Let that sink in for a moment. This is content they should have learned four years earlier. Even worse, only 24% of eighth graders were able to figure out that the sum of 7/8 and 12/13 is close to 2. What’s the take home lesson from all of this? The strategies we have traditionally used to teach fractions do not work. We need to take a different approach.
HOW HAVE FRACTIONS BEEN TAUGHT?
Just think back to your childhood. You’re in your 3rd grade classroom and you are starting a new unit called fractions. What are the images that come to mind? Pieces of a pie? Pizza?
But guess what? Leading researchers and mathematics experts, including Dr. Nancy Jordan from the University of Delaware and Dr. Hung-Hsi Wu from Berkeley strongly recommend against the pie model. They have recommended using the number line as a better way to teach fractions because it offers a more extendible understanding of fractions. The length model also lends itself to more real world applications (construction, architecture, sewing, manufacturing).
The pie model tends to reinforce the common misconception that a fraction is usually less than one. In reality, a fraction can be almost any point on a number line: 6/1, 24 ½ , 0/2, -¾ etc. Another enduring problem with the pie metaphor is that it makes fractions seem fundamentally different from other numbers. If students’ mental images of a fraction are wedge-shaped, what does it then mean to add, subtract, multiply and divide that wedge by other wedges? What is ½ of a pie divided by ¼? There are very few practical applications of fractions operations that involve circular shapes.
It’s much more useful to think about fractions on a number line. Take the tricky concept of dividing by a fraction. Using the measurement model of division with whole numbers, you can think about 16 ÷ 4 as how many 4s you can fit in the length 16. Using that same model of division with fractions, ½ ÷ ¼ , you can think of how many ¼-length pieces fit inside ½. You can probably even think of a circumstance in real life when you’d need to divide by a fraction. If you have a ½ yard of fabric and you need to divide it evenly into ¼ yard pieces, how many pieces could you make?
To address the deficit in fractions knowledge, we applied for and received grants from the National Science Foundation to develop a suite of fractions apps that focus on a length model (number lines and bars). We recently released our first fractions app, Fractions Boost EDU, which helps students develop a linear understanding of fractions through an engaging 3D racing game. When students need support in estimating a fraction, a 2D dashboard helps students see the meaning of the numerator and denominator. The phrase “tap __ times” surrounds the denominator, and as students tap, they see the fraction bar change from a whole to halves, thirds, fourths, etc. If they need further support, the phrase “move __ segments” surrounds the numerator, and they can see the length increase as they move the slider.
Through the game’s twelve levels, students learn to estimate, compare, and determine equivalence of fractions, while strengthening their understanding of fractions as a length or magnitude (the term often used in research).
Check out these research studies to learn more about fractions on a number line:
Jordan, N. C., Rodrigues, J., Hansen, N., & Resnick, I. (2017). Fraction development in children: Importance of building numerical magnitude understanding. In Acquisition of complex arithmetic skills and higher-order mathematics concepts (pp. 125-140). Academic Press.
Wu, H. (2008). Fractions, decimals, and rational numbers. Berkeley, CA: Author.