We explain steps in great detail to students but often omit the underlying concept. The topic of adding or subtracting fractions with unlike denominators is an example of this.
The example above right is a short cut for what is shown above left. These short cuts, which math teachers love to use, add to the student’s confusion because these rules require the student to use rote memorization which does is not readily retained in the brain.
I suggest using what I call a meaning making approach. I present the student 2 slices of pizza (images courtesy of Pizza Fractions Game) and explain the following setting. “You and I both paid for pizza and this (below) is what we have left. You can have the pizza slice on the left and I will have the pizza slice on the right. Is that OK?” The student intuitively understands that it is not because the slices are different sizes. I then explain that when we add fractions we are adding pizza slices so the slices need to be the same.
I then cut the half slice into fourths and explain that all the slices are the same size so we can now add them. Then the multiplying the top and bottom by 2 makes more sense.
Fractions is one of the most challenging math topics. Many high school and college students struggle to some degree with fractions. The Common Core of State Standards (CCSS), despite all the criticism, includes components to address the conceptual understanding of fractions. Below is a photo showing a 4th grade Common Core standard regarding fractions along with an objective for a class lesson I taught at an elementary school in my district. I subsequently presented on this at the national CEC conference in 2014. Notice the bold font at the bottom, ¨justify…using a visual fraction model.¨ The photo above shows an example of a model I used in class.
The photo below shows a handout I used in the lesson. The first activity involved having students create a Lego representation of given fractions. These would eventually lead to the photo at the top with students comparing fractions using Legos. The students were to create the Lego model, draw a picture version of the model then show my co-teacher or I so we could sign off to indicate the student had created the Lego model.
The Lego model is the concrete representation in CRA. In this lesson I subsequently had students use fractions trips (on a handout) and then number lines – see photos below.