I began writing this blog post after receiving an email from a teacher who wanted a copy of several of the games I used during a demonstration lesson involving fractional concepts. Not wanting to just give her “fish” I thought about how to give her more. Since receiving that email, I have rewritten the rough draft of this post many, many times. The first version of this blog post was way too long and I was pretty sure no one would read all of it. The second version was too short and would have only skimmed the surface.

Hopefully, this post is “just right.” I’ve included highlights from the research and support for the concrete development of fractional concepts, examples of concrete explorations of challenging fractional concepts, and games to make learning fractions fun.

“Clearly, the way fractions are taught must be improved. Because of the complexity of fraction concepts, more time should be allocated in the curriculum for developing students’ understanding of fractions. But just more time is not sufficient to improve understanding; the emphasis of instruction should also shift from the development of algorithms for performing operations on fractions to the development of a quantitative understanding of fractions.”     -Bezuk and Cramer

The above quote is from “Teaching about Fractions: What, When, and How?” by Nadine Bezuk and Kathleen Cramer. They wrote about the need for our schools to teach fractional concepts differently. They “encourage teachers to use an instructional approach that emphasizes student involvement, the use of manipulatives, and the development of understanding before beginning work with formal symbols and operations.”

Bezuk and Cramer make the following general recommendations for the teaching of fractions:

1. The use of manipulatives is crucial in developing students’ understanding of fraction ideas. Manipulatives help students construct mental referents that enable them to perform fraction tasks meaningfully. Therefore, manipulatives should be used at each grade level to introduce all components of the curriculum on fractions.
2. The proper development of concepts and relationships among fractions is essential if students are to perform and understand operations on fractions. The majority of instructional time before grade 6 should be devoted to developing these important notions.
3. Operations on fractions should be delayed until concepts and the ideas of the order and equivalence of fractions are firmly established. Delaying work with operations will allow the necessary time for work on concepts.
4. The size of denominators used in computational exercises should be limited to the numbers 12 and below.

Their article contains a gold mine of useful strategies and the reasoning for the use of these strategies. It would be worthy of your time to read more.
http://www.cehd.umn.edu/rationalnumberproject/89_1.html

One of the most challenging fractional concepts for me was fractional parts of a number. I knew the algorithmic procedure for finding the answer, but I was always the student who wanted to know “why” not just how. After having concrete experiences with fractions in my math methods course in college, I determined to always use these kinds of concrete experiences with my students. The following activity, perfect for the Teacher Time Station of The Tabor Rotation Framework, is amazing to watch and highly successful with students!

Each pair of students is given a set of miniature shirts cut out of fabric and a bag of interesting buttons. First, the students explore fractional parts of a set by pulling out two different kinds of shirts for a total of 4 shirts. The teacher asks students to talk about their sets in fractional language. An example of a student response might be, “I have 2 red bandana shirts and 2 red plaid shirts. That means that 2/4 of my shirts are red plaid. If I separate the shirts like this you can also see that 2/4 is equivalent to 1/2 because it’s obvious that 1/2 of my shirts are red plaid and 1/2 are red bandana.”

Next, the teacher asks the pairs of students to select 12 buttons from their bags and arrange them in equal numbers of buttons on the shirts. The students are asked to talk about their buttons in the same way they talked about their shirts. After a few variations on numbers of shirts and number of buttons, the teacher encourages the students to read the following question with their partner and determine the answer. “What is 1/4 of 12? How do you know?”

Almost every single student will look up and around and start trying to “think” about how to answer the question. The teacher should suggest that they look down and use the shirts and buttons to help them answer the question.

It won’t take long for one of the students to push a shirt up and say, “It’s 3!” The explanation of “how they know that” might be, “We pushed up 1/4 of the shirts to show 1/4, so wouldn’t 1/4 of the 12 buttons be on the same 1/4 shirt?”

The lesson continues with different numbers of shirts, different numbers of buttons, and deepening understanding of fractional parts of a set and number. The teacher may write the abstract representation for what the students are doing concretely, but that is not the emphasis of the lesson.

Since we know the brain learns best concrete-to pictorial-to abstract, the following technology can be used to bridge the conceptual development from the concrete buttons and shirts to the pictorial. The following week a teacher might put a highly interactive app, like Educreations, (http://itunes.apple.com/us/app/educreations-interactive-whiteboard/id478617061?mt=8) in the Application/Technology Station of the Tabor Rotation.

The students are instructed to “teach another student, in another state, about fractional parts of a set and number using the interactive white board.” This lesson is posted online and can be viewed by the entire class. It can also be used as a performance-based assessment. The following link is an example of one explanation of fractional parts of a set and a number.
http://www.educreations.com/lesson/embed/562151/?ref=app

If you have fraction bars and want to help your students explore fractional parts of a whole, equivalency of fractions, and readiness for addition/subtraction of fractions with like denominators, then the following games would be extremely useful.

Fraction Flip, Make Mine Whole, Equivalent Concentration

If you’d like to continue your exploration of fractions, then you may want to visit the following sites.

NSA has a 32-page booklet full of pre-assessments, activities, games and really engaging fraction lessons.
http://www.nsa.gov/academia/_files/collected_learning/elementary/fractions/What_Fraction_am_I.pdf

Kids R Kings Katering is a performance-based assessment task developed by Mary Benton and Renee Patrick. This real-world application of fractions and decimals would be an incredible experience and connection-making tool for 5th-7th graders.
http://www.nsa.gov/academia/_files/collected_learning/elementary/fractions/kids_r_kings_katering.pdf

To learn more about possible challenges to a student’s understanding of fractions, read this practice guide from the U. S. Department of Education. It includes actual student examples, recommendations, and “roadblocks” with ways to get around these roadblocks.
http://ies.ed.gov/ncee/wwc/pdf/practice_guides/fractions_pg_093010.pdf#page=25

Why use the activities and games in this blog? Why do research to become more effective in your instruction of fractions? If a shirt and a bag of buttons could change a student’s mathematical high school career, wouldn’t it be worth it?

“We shall not cease from exploration and the end of all our exploring will be to arrive where we started and know the place for the first time.” —  T.S. Eliot, Four Quartets