On a recent site visit, I was observing in a fourth grade classroom. The teacher started the lesson (Unit 6, Session 2.1) by writing “3/2” on the board and asking students to name the fraction. Most said “three halves” although one or two said “two thirds.” The teacher then displayed two blank 4 x 6 rectangles. She established that one rectangle was the whole, and asked students to use their copy of the rectangles to draw a representation that showed 3/2.

The math coach called me over. “You have to talk to Lamar!”

I walked over and asked Lamar what he was thinking.

“It’s a defective fraction. You can’t have more than one. You just can’t.”

I asked, “Well, let’s try drawing it on these rectangles. Can you do that?”

Lamar shaded in one whole rectangle, and said, “That’s two halves. That’s ok. But there’s no more left!”

“Well, what if you shaded in half of the other rectangle?”

Lamar shaded in 1/2 of the second rectangle and looked at it for a while. Eventually I asked, “So if we wanted to name the fraction you’ve shaded, what would it be?”

“Well, it’s three halves. Maybe it’s not defective!”

I had to smile to myself. Lamar called them “defective” which isn’t all that different than what these types of fractions were called when I was a student and a beginning teacher, when we called them improper fractions. There is nothing improper or defective about them.

I was curious about the students who read the fraction as 2/3, as well as Lamar’s statement about 3/2 being a “defective” fraction. What is their understanding of fractions? Do they think 2/3 and 3/2 are the same thing? What do they understand about fractions greater than 1?

After a brief whole class discussion about 3/2, the teacher had students start making fraction cards—a deck of 40 cards that students create, with representations of the given fractions, and used to compare fractions and create a class number line.

As I wandered around the room, watching and talking to students, I kept my earlier questions in mind.

I joined the group of 4 boys that included Lamar. He was making a fraction card for 2/4. I asked if he knew of any fractions that were equivalent to 2/4.

“Sure, 4/8.”

Another boy in the group chimed in, “There are a lot of fractions that are equivalent to 2/4.”

“How many do you think there are?” I asked. The four boys agreed there would be a lot.

I asked, “Do you know what infinite means?”

Lamar replied. “Hmm, I think so. Like it goes on and on and on. You mean like infinity?”

“Yes. Do you think there are an infinite number of fractions equivalent to 1/2?” The boys looked at each other, and one finally said, “Hmm, maybe.” Another added, “I’m not sure it would be infinite.”

I asked a few more questions. “Well what if the denominator was 100? … 800? … 1,000,000?” The boys answered each of those correctly, although for one million Lamar said, “5,000. No! 500,000.”

“What if the denominator was 5?” There was a bit of a silence, then a boy chimed in. “2.5!” I told them we normally don’t write decimals as part of fractions, but we could write the numerator as two and one half. Afterwards, I realized I should have asked the group what the numerator would be if the denominator was 1/2; I wondered if Lamar would think was a defective fraction.

As always, I was fascinated by students’ work and talking to them about their mathematical ideas. I find such conversations particularly interesting when students are entering into a new realm of mathematics. In the case of Lamar and his classmates it was rational numbers, but it’s also true when students start working with larger numbers or move from one operation to another. They bring a great deal of mathematical knowledge to this work and it’s interesting to watch as they try to take their existing knowledge and fit it into new mathematical frameworks.

### Keith Cochran

*Investigations*Center for Curriculum and Professional Development. He co-directed the development of

*Investigations 3*, was a senior author of the 2nd edition, and has extensive experience providing professional development for teachers, schools, and districts implementing Investigations. Keith taught in a range of settings, including Native American Nations (AZ) and the Clark County School District (Las Vegas, NV), where he taught grade 5 and became a Math coach.

#### Latest posts by Keith Cochran (see all)

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We are constantly struggling with the question of differentiation in math classes. In my school we have done away with the fixed mindset model of pullout classes for either the ‘top’ or the ‘bottom’ students in favor of heterogeneous grouping and a growth mindset model. All students can learn math; each student must construct their own understandings; there is power in the social/collaborative work.

(See Carol Dweck’s work on Mindset. A good place to start is Po Bronson’s article in New York Magazine: How Not to Talk to Your Kids.

http://nymag.com/news/features/27840/

See also the Making Learning Visible work of the Project Zero Group: “Powerful learning is purposeful, social, emotional, empowering, and representational.”

http://www.pz.harvard.edu/resources/visible-learners-promoting-reggio-inspired-approaches-in-all-schools)

It is the teaching moments such as you describe in which the right questions at the right moments stretch the math in directions that allow students to explore mathematics beyond (or before) the focus of the lesson at hand.

My question is how might we capitalize on those teaching moments to intentionally and consistently differentiate instruction? Certainly these ideas can be part of the math discussions following the exploration phase of lessons, but how much and how far do we take this part of the discussion? …and having these explorations roll over into differentiated homework is another question.

Thanks for your response, Dan. Differentiation often comes up when we lead professional development, and we emphasize the importance of teacher content knowledge. In order to meaningfully differentiate instruction we believe teachers need to know the math, know the student, and know the curriculum. There are numerous supports in Investigations to help teachers with this: the Mathematics in This Unit; the math focus points for sessions and discussions; Ongoing Assessment questions; suggestions for differentiation in many of the sessions; extended differentiation activities at the end of each Investigation; and an overview of differentiation in the Implementation Guide at each grade level. Each of these components is professional development for the teacher. It’s also important teachers know the curriculum. Quite often “extensions” or “interventions” (particularly in the case of misconceptions) are addressed in subsequent lessons.