Fractions Playlist

It's early in the year (week 2) in Navy Band's Math Block, and these 2nd and 3rd graders are growing their understanding of fractions. In this learning episode, we join Navy Band, a group of 2nd and 3rd graders in their first months of learning at Long-View. Their teacher, Kevin Moore, draws a vertical number line (one of the important models for thinking that we often use with young mathematicians) on the board. He places 0 about a ⅓ of the way up the number line. He then writes two numbers, two and ½, on the board. He asks the community, “where would these numbers go on the number line?”. 

One learner comes forward to place the points, placing 2 at the top and ½ above 2. Kevin follows the learners thinking, ensuring the entire community is considering this. The community says that they disagree, Kevin remains neutral, and he inquires about their disagreement, pressing learners to explain why they perhaps disagree. The learners continue to offer other ideas of where they believe the numbers belong on the number line. As they work, their teacher draws out further explanations, and through the discourse, learners grapple with and transform their understanding of number. As you watch, you may notice how Kevin facilitates a discussion, where many voices are heard and understanding is constructed together. This early Concept Study illustrates the rich discussions that can lead to powerful understandings of fractions.

As the Concept Study from the previous video picks up in this second clip, there are many important teaching moves to notice. Kevin utilizes the number line as a model for thinking and then moves quickly away from this model, writing a fractional count on the board, 16(⅛), with the expectation that learners are working to grow an idea and make connections with the prompts previously explored. The model is not meant to be used again and again. Instead, these models should be internalized and used as a structure for thinking. For Kevin to guide learners in doing this type of thinking, you will notice that he makes references back to prior prompts; rather than answering for learners, he presses them to transfer or utilize information gleaned to unlock new understanding. A second crucial teaching move you may notice is the use of precise language expectations. Precise language is crucial in understanding a concept, leading to clearer thinking. Kevin does not simply fill in the blank for learners or offer "I know what you meant to say…" instead, he presses learners to do the cognitive work on their own.

We join Navy Band again, later during the same Math Block as they continue to weave together threads that will lead to the foundational understanding of fractions that other concepts are built. This time, Kevin presents learners with an area model, being very deliberate in how he speaks and notates the thinking. The model begins the conversation, and again you will notice that Kevin moves quickly away from the model, expecting learners to work to internalize it. Learners are thinking relationally, considering the relationships between numbers and leveraging those relationships to reason with number. As you watch, you may also notice the struggle, and struggle is good. When learning is at the center, the challenge is embraced, and we see the struggle as an opportunity to learn. As these mathematicians work, Kevin provides feedback not only about the mathematics, but about learning in community, "I don't want to know that you know it. I want you also to help others who don't know it."

We again join Navy Band a week after the Math Block from the previous videos. They know that fractional parts of a whole must be equal in size, and they have been exposed to the measurement model of fractions (i.e., they have seen fractions on a number line). They have had more experiences seeing fractions depicted on a number line and fewer experiences seeing the more traditional area model of a fraction as a shaded part of a whole. This is purposeful, as we emphasize the measurement model and do not over-represent the shaded area model that we too often see classrooms overly rely on. The children in this band also understand how many fractional parts make up a whole (i.e. they know 4 fourths equals a whole and 5 fifths equals a whole and so on).

Kevin begins the Concept Study on this day with a series of questions to be sure that the children see 6/7 and 6(1/7) as equal, emphasizing that when we make the unit fraction clear, as in the notation 6(1/7), we can more readily understand what is being counted. In this case, we are counting one-sevenths and we have six of them.

He then moves to scaffolding the kids toward figuring out how to find the number of wholes in an improper fraction, such as 21(1/7). Dasher attempts to make sense of this using his knowledge of 7(1/7) as equal to one whole, and he spends a few moments struggling to find the language to say "Three seven sevenths is equivalent to twenty-one sevenths." (Note that Kevin does not fix Dasher's inaccurate language for him and Kevin also does not give Dasher a pass by saying something like, "I think I know what you are trying to say." Instead, he continues to push Dasher to make his thinking more clear and thus his language clearer by asking questions or disagreeing with him.) Once Dasher is able to articulate that 3[ 7(1/7)] = 21 (1/7), Kevin scaffolds the band toward a conclusion that 21(1/7) = 3(1).

Juliet then makes a comment that she "doesn't understand any of it." Kevin then leads her (and likely many of the other kids) back through the reasoning that led to this conclusion, supporting Juliet in thinking about units of seven-one-sevenths, 7(1/7), to conclude that 21 (1/7) = 3 wholes. Ultimately, when Juliet seems to now understand, Kevin gives her feedback that he appreciated that she "took ownership of her own learning by asking the question" but also presses her to be more specific next time with her question, telling him more specifically what she does and doesn't understand.

As a teacher, the crucial idea to understand here is that we want our learners to experience fractions as “numbers.” That is to say, we do not want them to think of the fraction ¾, as so many children do, as merely a numerator of 3 and a denominator of 4. Children can often shade ¾ on an area model but not understand it as a value in relationship to other numbers. A fraction is not simply a mental image of the area model. In this video you see Navy Band have an experience that continues to build them toward understanding fractions as a number, as they work with 21(1/7) in a similar fashion to how they might if it were just 21, decomposing it into smaller units of seven sevenths and ultimately understanding it as equivalent to 3 wholes.