Composite Units and Fractions

In my previous blog, I talked about a composite unit, what it is and how it plays an important role in many different aspects of students’ construction of mathematics. One of these areas is fractions. So how does the student’s ability to take a number as something that is countable affect their understanding of fractions? It begins with the simple understanding of sharing equally, and progresses into number sequences with a unit of 1/n.

Think about a candy bar. What would a piece look like if five people were sharing the candy bar equally. The way students do this will differ depending on whether or not they have constructed a composite unit. Some kids will mark off a piece, and then make four more pieces the same size to see if they take up the whole bar, and start over if they do not. A student with a composite unit will be able to “project” five equal parts onto the bar simultaneously, and will often be fairly accurate with their first attempt. The composite unit makes these sharing tasks much more accessible for these students.

Eventually, we want students to generalize their number sequence. What I mean by this is we want them to see any number as a possible unit from which to build a number sequence. One isn’t the only number by which we can count, but neither are 2, 3, 5 and 10. We see the importance of this skill and teach kids to count by two’s, and especially by tens. Think how powerful it would be for a student to realize that they can construct a number sequence of fourths. If you are an elementary teacher, you have probably come across children struggling to make sense of improper fractions. When students come to understand 1/4 as a unit that is iterated 5 times to make 5/4, it opens new possibilities for conceptual understanding of operations with fractions. This understanding is very similar to seeing the number 5 as the iteration of one, five times on the candy bar. Can you see the connection?

The development of a composite unit contributes to their ability to come to know fractions. The concepts developed with whole numbers lend themselves to making sense of improper fractions, as well as operating with these numbers. We already saw how the composite unit is important for additive as well as multiplicative thinking in previous blog posts.  With the added importance it plays in fractional knowledge, I would say that the construction of a composite unit is a very important goal that should be a focus of mathematics instruction for all children.

Meet the Children Where They Are: Episode 4

Narrator:  “Although this series makes this interaction seem like a long period of time, it was actually only about 3 minutes…well, maybe 5!  Time flies when you’re having fun, okay?  Anyway, can Bob count two hidden piles of rocks?  Let’s get back to the action.” (Math research associate narrates…) RA:  So it was the bottom… Continue Reading

Episode 28 | Pedagogical Content Knowledge: Its Role in Teaching

Joining Chris in the studio this week is Paul Reimer, a Sr. Researcher at the AIMS Center working with our Early Mathematics studies. Paul is also a student in the Michigan State University Doctoral Program, studying the effects of teacher beliefs on student learning. We discuss his studies and how they connect with our work here at the AIMS Center.

Continue Reading

A Heart for Professional Learning

Who do you rely on professionally? I could name a long list of people, places, journals, periodicals, podcasts, and websites, but most recently I listened to my colleague Chris Brownell’s recent podcast with Director of Special Education Studies at Fresno Pacific University, Megan Chaney. Megan is doing her doctoral research on teachers dispositions and she… Continue Reading

“Play is the Answer to how Anything New Comes About” – Piaget

As the early math team moves forward on the work we are doing, the concept of practicality is an issue we are addressing. One-on-one interviews with the children have taught us a wealth of information about young children’s mathematics, but it is not a realistic structure that early childhood teachers have time to do in… Continue Reading

Which is Bigger?

Which is bigger 5/6 or 7/8? If the answer isn’t popping into your head in seconds, you are not alone. Fractions are one of the most misunderstood concepts among both young and old in mathematics.  They don’t seem to follow the same rules as whole numbers.  Many of us purposely never work with fractions at… Continue Reading

Episode 27 | Professional Noticing Part 3: Deciding

In the studio with David Pearce and Wilma Hashimoto two of the AIMS Center’s Research Associates, and we discuss the third aspect of Professional Noticing: Deciding. We discuss how this aspect takes place in the midst of classroom activity, and how it is dependent upon the two prior aspects of: Attend and Interpret. We end up discussing how this supports the goal of creating a student centered classroom, one in which the learner’s thinking and conceptualization is valued as the starting place for academic learning.

Continue Reading

Change Unknown

Recently, while working with students, we offered up a situation where nineteen counters were placed under a cloth.  Seven of the counters were pulled out and the students were asked how many remained under the cloth. One child extended ten fingers, pulled them back, and then re-extended nine. He pulled back seven fingers, one by… Continue Reading