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A few weeks ago, my niece started a unit on fractions with her class. Every time she had a comparison problem, such as comparing ½ with ¾, she would say something like ¾ are greater than ½.
After hearing comparisons like this a few more times, I decided to ask her this question:
What is a fraction? What would your answer to this question be?
What do you think her answer was?...
She told me that a fraction is something that has a top number and a bottom number.
This is the response that I have also heard from my years in the classroom and from working with children in my role as a math coach. Maybe this is a definition that you often hear too.
When I was growing up, I used this language of “top number” and “bottom number” too - because this is how I was taught. And when I first started teaching, I continued to use that language- not because it literally is a “number on top”, but honestly because it was easier for me to just say that in order to indicate where I was talking about when talking about a numerator, even though it isn’t technically true. I hope you’re reflecting on your own practice right now to think about whether you as an educator, or a parent, have used this phrasing of “number on top” or “bottom number” before. Considering that “a high percentage of U.S. students lack conceptual understanding of fractions, even after studying fraction for several years (Siegler et al., 2010, p. 6, as cited in Van de Walle et al., 2014, p. 202), chances are that many you at some point have used some of this phrasing.
But if it isn’t true, then we shouldn’t be saying it.
Children take us literally and at our word - which means we have to be extremely intentional with our words.
This Fractions series is on a mission to deepen our understanding of fractions so that we understand exactly why phrasing such as “top number and bottom number” can hinder a child’s understanding, as well as what to say and do instead.
In Part 1 of this series we talked about the 5 mental actions that are needed in order to have a deep understanding of fractions. But, what exactly is a fraction? Children and adults alike often have different interpretations of how to describe a fraction. In this episode, Part 2 of the series, we define the term and also discuss instructional moves for all elementary teachers, kindergarten through 5th grade, that lead to a deeper understanding of fractions. Be sure to pause right now and press that subscribe button to know when the next episodes about fractions understanding are released.
To start, let’s go back to the definition of a fraction that my niece gave:
A fraction is something that has a top number and a bottom number.
First and foremost, a fraction is a number, not a “something”. The word something suggests that fractions are separate in children’s eyes - just a random part of math that is talked about near the end of the school year.
According to the text, Teaching Student Centered Mathematics, “fractions are numbers with special names that tell how many parts of that size are needed to make the whole” (Van de Walle et al. 2014, p. 202). Did you hear that important phrase- of that size? Oftentimes when fractions instruction begins, this important relationship is left out, which opens space for us to slice circles and rectangles and ask gathering information questions such as, “how many are shaded in?” and “how many pieces do we have altogether?” I’ve seen countless worksheets and journal pages with this very wording - so what’s the big deal? It’s a valid question right?
It is valid. These questions, however, are basic. They are only collecting information about whether a child is essentially able to count one to one and then mimic the teacher’s explanation of how to write in fraction notation. When we ask children questions such as these, we are not empowering them as learners - we are actually inviting them to disengage from authentic thinking and learning.
These questions also lead to language such as “two over six” for example, which makes it sound like each component of the fraction is a whole number. When referring to fraction notation and then saying two over six, instead of 2/6 of the whole adds to a child's confusion about the definition of a fraction. We must do better than this.
Thinking through this helped me understand why my niece was saying ¾ are greater than ½. The concept image she has formed and is beginning to internalize, emphasizes procedures and the abstract fraction notation and incorrectly reasons that 3 and 4 in the number ¾ are in fact two separate numbers. When viewing ¾ in this way, it makes sense to use the word “are” instead of “is” when comparing fractions.
In episode 6 of the podcast, Creating Powerful Visual Images, we learned that once a visual image of a concept has been built, it becomes increasingly difficult to change as a child gets older.
We have to begin to shift our instructional moves in order to build these powerful visual images that help children create connections in their minds and think about relationships.
But where do we start? What instructional moves should we be using to tap into children's funds of knowledge about fractions and to ensure a deeper understanding?
Children, even those in lower elementary, need to understand that a fraction is a number - it represents a quantity. When written in number form, a fraction has three key components: a part of the whole that is written above the fraction bar, the fraction bar itself, and a part representing the whole that is below the fraction bar. Sometimes these components are referred to as elements of a fraction (Whitenack, Cavey, & Henney, 2015; Van de Walle et al., 2014).
But before moving to this abstract fraction notation, children need experience with concrete representations, and also pictorial representations in order to be flexible thinkers about fractions.
The text, Extending Children’s Mathematics Fraction and Decimals, gives us some guidance when it states that “what really makes a fraction is [that it is] a number whose value is determined by the multiplicative relationship between the numerator and the denominator” (Empson & Levi, 2011, p. xxii).
In other words, we can think about a fraction, let’s say 2/6, and the fact that this is 2 times what you would get when dividing a whole into 6 parts (Van de Walle et al., 2014).
This is complicated thinking and we need to give children many different experiences, along with time to think, feedback, and discourse opportunities in order to develop this relational thinking.
Research suggests that beginning fraction instruction by introducing Equal Sharing Tasks helps us reach these goals. This will also help us get a better sense of how children are thinking about fractions and partitioning and fragmenting, one of the 5 mental actions discussed in episode 20.
Examples of Equal Sharing tasks include the following:
2 friends want to share 5 cookies so that everyone gets the same amount.
And
4 friends want to share 10 brownies so that everyone gets the same amount.
When children are thinking deeply about partitioning, or breaking into equal size parts, and thinking about a given part of that whole, they might not automatically know how to write out that number in fraction notation. The point here is that this is okay! Just like we shouldn’t rush children from the concrete directly to the abstract, we shouldn’t rush them toward writing out fractions before they are ready. As Empson and Levi state (2011),
“Fraction terminology is not part of children’s intuitive knowledge of fractions” (Empson & Levi, 2011, p. 24).
Equal Sharing Tasks add meaningful context for children and afford them the opportunity to draw pictures to help them make sense of the mathematics that is happening.
Without the formal language of fractions, children can still think about problems that have solutions involving fractional parts, such as Equal Sharing problems, and they are very capable of “represent[ing] their solutions without using fraction terms or symbols to describe the final share” (Empson & Levi, 2011, p. 24).
So again, if a child is able to partition a rectangle into halves, or thirds, and can talk about it, but can not yet write out ½ or ⅓, they are not in need of intervention or pages and pages of additional worksheet practice that has them writing down fraction notation.
In person, children can use blank paper, a whiteboard and marker, or some chart paper. When thinking about online options, Google Jamboard is a fantastic option because it has a drawing tool, along with Classkick, or another online whiteboard tool. All of these options will make it easier for children to share their thinking with classmates as well. These Equal Sharing tasks also lead to richer discourse and the opportunity to replace the basic question of “how many are shaded in?” with one that probes thinking, such as:
“How many of these parts fit into the whole?”
This type of question “focuses on the size of the part relative to the whole” and helps children understand that they have to think about relationship between the “size of the fractional part and the whole [in order to] determine the value of the fraction” (Empson & Levi, 2011, p. 24). Once children have had time and space to think deeply about the relationship that is happening when we talk about fractions, we can move on to talking about how to write a fraction in fraction notation. What we can do as educators and parents, can write this notation ourselves when talking with children about their work to show another representation while at the same time not expecting children to mimic our actions.
By the way, if you don’t already have this Extending Children’s Mathematics Fractions and Decimals book- I highly recommend it! The authors give easy to implement suggestions and write in user-friendly language for parent and educators. I purchased this book a few weeks ago at the suggestion of a Twitter friend and it is seriously a must have for your professional library.
ANNOUNCEMENT of 3rd GIVEAWAY WINNER
Before we go, I want to announce the third winner of the latest Kids Math Talk Giveaway! We had 4 entries and congratulations to stephbsinger who writes - “I’m so glad I found Kids Math Talk a few weeks ago. I’ve listened to three episodes and have greatly enjoyed the interviews that Desiree conducts, as as a variety of teacher tips and insights.” Thank you so much for that review and be sure to email me at kidsmathtalk@gmail.com so that I can send you your free 5 Practices in Practice book!
CALL TO ACTION
When we talk about mathematics together, when we keep the conversation active and positive - that is when we grow in our understanding and learn how to better support our students.
Teachers- I have two challenges for you this week. The first is to talk to one of your colleagues about the definition of a fraction. The second challenge is to try out one of these equal sharing tasks with your students.
And parents - Talk to your child’s teacher and ask them how they are using equal sharing tasks to help children access their funds of knowledge and build flexible thinking about fractions. I also challenge you to try out one equal sharing task with your child.
And then whether you are a parent or educator - I challenge you to take a picture of this thinking and then tag me on Twitter, along with the hashtag #kidsmathtalk or post it in the Kids Math Talk Facebook group so that we can continue to learn together and keep this conversation about math going strong on social media.
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