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We use fractions in our daily lives, like in cooking, we see them on nutrition labels, and they are a part of many sports, and yet fractions can be an uncomfortable topic for a lot of adults and children. But regardless of how we feel about fractions, educators are either already teaching a unit about fractions or will soon be. And for those of us who are parents, chances are that your children are learning about fractions right now.
I added this week’s episode into the rotation because I am currently taking a Fractions Math Recovery course that is completely changing my view of fractions and I want to share my new learning with you.
The U.S. Math Recovery Council offers professional development courses that truly transform your thinking and teaching and I highly recommend checking out all of their offerings. Their url address is www.mathrecovery.org.
They believe what we at Kids Math Talk believe - that kids should have time and space to learn concepts and should have the chance to talk about math and demonstrate understanding through performance tasks, not just paper and pencil activities. And all of their courses, like this fractions course, are extremely well researched. I’d like to give a special shout out and thanks to the members of my Fractions cohort and our teacher Julie B. for helping me to unpack the information that I am going to share with you today.
What follows is not an exhaustive explanation of fractions - This episode is focused on giving educators and parents some “look fors” and questions to ask to help children begin to explore their understanding of fractions as numbers.
And to do this I’ll begin with a question:
How many different mental actions do you think happen in order for someone to have a deep understanding of fractions as numbers?
Turns out there are 5 - unitizing, fragmenting and partitioning, iterating, disembedding, and distributing. Ever heard of these before? I know I have read many books about math education that included notes about partitioning and iterating, but I never knew about all five of these or the way that they progress in order to create fractions understanding before being introduced to them in this Math Recovery course. So let me introduce to these to you.
The first of these mental actions is called Unitizing.
Unitizing is known as treating a quantity as a single unit. We can ultimately make any quantity into a unit. For instance, we might say that we have 12 eggs, but we can reorganize this and think about having 1 dozen eggs. The unit thus switched from units of 1 egg, to units of 12 eggs. Children need to have time to explore and understanding this mental action with whole numbers in order to then reorganize their thinking and apply this mental action to fractions.
When first learning about this mental action in the course, I kept thinking about Counting Collections with whole numbers. This is a super engaging activity for children and a great way for adults to observe and see how children are thinking about units and grouping items. You can use anything to create a counting collection - those tiny erasers that kids love work really well. Simply gather a random amount of them, place them on a table or even on the floor, and then ask “how can you figure out how many are here?” and then watch how
children organize and group the items. Having about 50-100 items for the first time is a good start. You can also do a Google search for Counting Collections for more ideas.
The second mental action is known as fragmenting and partitioning.
My introduction to partitioning comes from The Teaching Student-Centered Mathematics (2014) book that talks about partitioning as “sectioning a whole into equal-sized pieces” and that it is a “major part of developing fraction concepts” ( Van de Walle et al., 2014, p. 211), but this course has equipped me with some questions to ask during student observation and also when debriefing with teachers.
One important question to ask to begin to assess a child’s understanding of this mental action is:
Is the child able to use the entire whole when breaking it up into pieces?
That is to say- do they recognize that every single portion needs to be accounted for or will they simply ignore any part that is not making sense to them?
Another question to ask is-
Is the child thinking about equal size pieces?
When working with children on creating equal shares, it is helpful to use language such as “1 out of 3” as opposed to ⅓ so that the language matches the action
Parents - you can think about these same questions when working with your child at home or you can even ask these questions during the next parent-teacher conference to get the conversation about fractions started.
Once children understand unitizing, fragmenting and partitioning, the next mental action to develop is that of iterating.
Iterating is “the counting or repeating [of] a piece” (Van de Walle, 2014, p. 213). This mental action is more easily understood when thinking about measurement, such as repeating units of ⅕ on a number line.
One activity to strengthen this mental action is to count fractions with children. Oftentimes we don’t think to do this, but this will help reinforce the fact that fractions are numbers and that they can be counted. Just like whole numbers can be counted.
For instance, ⅕, ⅖, ⅗, ⅘, 5/5. This will help children begin to make the connection that ⅖ is the same amount as ⅕ plus another ⅕. Having a visual representation for this measurement model will be helpful as well.
1st 5 Practices in Practice Book Giveaway Winner Announcement
Let’s take a quick break to announce the first winner of the latest Kids Math Talk Giveaway! We had 5 entries and congratulations to Lennyvermaas who is the first winner and writes - "Kids Math Talk is one of my favorite podcasts and is one that I listen to on a regular basis. Desiree interviews a wide variety of people that provide resources and strategies for elementary teachers to help students see the wonder, joy, and beauty of math. The visit with Peg Smith as she talks about the “5 practices to Improve your Practice” was another great episode. Always well worth the time to listen.” 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!
After iterating, disembedding is the 4th mental action.
We as educators and parents need to observe to make sure that children understand this action which “involves taking a part out of a whole without destroying the whole” (Hackenberg et al., 2016, p. 19). One quick activity to try in order to figure out if a child understands this disembedding is the following:
You will need two strips of paper, a pencil or marker, and a pair of scissors. The two strips of paper need to be the same size and length.
Take one of those strips and partition it into five or six equal parts. You can do this with the pencil or marker, just ensure that all of the portions are equal size.
Show both of these strips and ask the child you are observing to take the unmarked strip and cut off a set amount (⅖ for example)
Next, ask the child to tell you how much of the whole that cut portion represents. The answer in this case would be ⅖ of the whole.
Then also ask the child how much the uncut portion represents. The answer for this scenario is ⅗.
This Math Recovery course is helping me realize that this mental action is one that I have always just assumed that children were able to perform but a child who does not know how to disembed might try something like this: counting the 3 parts that are left along with the 5 parts of the uncut strip, for an answer of ⅜.
Having physical manipulatives such as fraction strips, Cuisenaire rods or virtual manipulatives from sites such as www.brainingcamp.com will help children visualize and develop this mental action.
This mental action is one that I am continuing to think about and read about myself.
The final mental action developed in order to have a deep understanding of fractions is called distributing.
Distributing requires a child to have flexibility with the other mental actions as well as an understanding of the multiple levels of units. While the word distributing might immediately bring in thoughts of multiplication, it is important to note that we will confuse children if we approach the multiplication of fractions in the exact same way that we approach the multiplication of whole numbers. With whole numbers, we can rely on the strategy of repeated addition. This is not going to work when multiplying fractions though. Instead, multiplying fractions involves taking a fraction - or a part of another fraction.
You might have noticed how I haven’t mentioned anything about adding or subtracting fractions. This Fractions Course has helped me realize that all of these mental actions need to be in place first in order for children to truly understand performing operations with fractions.
So if you’re not sure about committing to taking a course right now, you can still learn more through the book that goes with this course called Developing Fractions Knowledge. This book is great for not only teachers, interventionists, but also for parents to have because there are detailed descriptions of activities to try out with children to develop each mental action. You can grab this by going to us.corwin.com. Use our special code, KMTSHIP at checkout for free shipping.
CALL TO ACTION
This week just notice all of the fractions around you. Take a picture of the fractions that you and your child or that you and your class find or let me know how one of the activities mentioned in this episode went for you by tweeting me with the hashtag #kidsmathtalk
Let’s continue to learn together and keep this conversation about math active and positive.
References
Hackenberg, A., Norton, A., and Wright, R. (2016). Developing Fractions Knowledge. SAGE.
Van de Walle, J., Karp, K., Bay-Williams, J. (2014). Teaching Student-Centered Mathematics Vol. II : Developmentally Appropriate Instruction for Grades 3-5. Pearson.