Tag Archives: Fractions

I Almost Missed It

If you read the previous posts from the Coordinating Units team here at AIMS, you likely know that we are studying how children learn about fractions. Earlier this week I realized that I almost missed something amazing and encouraging about how much our students are actually learning. The tasks we are using with them are so simple in nature, easily administered, and engaging for the students that when the students were able to easily answer a question that I have always asked, I forgot how difficult that question had been for many of my former high school mathematics students. So why are these second and third grade students coming to this understanding of fractions when it has proven so difficult for students in my past experience?

Here is the question these elementary students responded to with clear understanding: Which is bigger, one person’s share if five are sharing a candy bar, or one person’s share if six people are sharing a candy bar? The students were given a strip of paper and scissors, and asked to first find the size of one person’s share if five people were going to share the bar. Once they had achieved this goal with precision and understanding, they were then asked to show the size of one person’s share if six people were to share the candy bar. As soon as they began to think about the task with six people, I would ask them if they thought this piece should be larger or smaller than when five people were sharing. While many of the children immediately knew that the piece should be smaller when six people shared than when five people shared, others knew after just a few experiences with each task. What’s the big deal, you ask?

Shortly, we will attach the number name of ⅕ or ⅙ to these pieces they are constructing. If you are in math education, whether elementary or secondary, you likely have experienced the difficulty students of all ages have in conceptualizing that ⅙ is smaller than ⅕ since 6 is bigger than 5. It is one of the reasons many educators say that whole number understanding gets in the way of fractional understanding. Dr. Les Steffe’s research implies that the opposite of this assumption is true. As it turns out, when we leverage their understanding of whole numbers in the construction of fractional understanding, the students can more readily make sense of their meaning. The ease with which these elementary students made sense of what used to seem like a difficult question for my high school students, I believe, corroborates Steffe’s hypothesis. You see, we used their understanding of the whole numbers five and six in order for them to explore what happens when you try to fit five and six into one whole and, voila, it just made sense to them.

I am so excited for the possibilities that we are learning about. Follow our blog to learn with us.

Sharing Can be Rigorous Work!

The term “rigor” has been highlighted in education since the Common Core Standards have been adopted. The Common Core Standards have been deemed to be more rigorous and, therefore, students should be engaged in more rigorous lessons. What does it look like for students to be engaged in a rigorous task? Especially in the K-2… Continue Reading

Unintended Opportunity

The team I work with at AIMS has begun working with students on fractional understanding and it has been an interesting couple of weeks. We are seeing the students have opportunities to learn things we never intended but still fit right in with expectations in elementary classrooms. Personally, I have discovered that when we develop… Continue Reading

Partitioning Shapes: Is it Geometry or Fractions?

Partitioning Shapes: Is it Geometry or Fractions?

How early should we teach words like half, thirds, and fourths to children? I know that I have often heard that we give young children things they are not developmentally ready for, and I agree. But when it comes to having language identify a concrete experience, I think children can handle it. I was measuring… Continue Reading

Finding Math in Unexpected Places

Finding Math in Unexpected Places

I was reading Inchworm and a Half with my 6-year-old daughter, Bethany, last night for the 40th time. She loves reading the section, “Squirmy, wormy, hoppity-hoop! We measure everything, loopity loop.” Even before she could read books she memorized this section and would “read” it. The book is about an inchworm that loves to measure… Continue Reading

The Problem of Anwar’s Camels

I just started reading Fractions in Realistic Mathematics Education by Leen Streefland, and there, on page 5, Streefland gives as an example an old puzzle problem that I remember giving my students more than 40 years ago. “An old Arab, Anwar his name, decreed before he died that his eldest son inherit one-half, his second… Continue Reading