Being Responsive to Student Thinking

When educators work with a partner or in small groups to examine and annotate student work, it allows for student thinking to be unpacked – their understandings, strategies and transitional conceptions. Here, student thinking from a Grade 3 classroom, a Grade 7/8 classroom and a Grade 9 classroom is shared for educators to use in their own professional learning, individually or school-wide.

Grade 3

In this Grade 3 classroom, students were given sets of pre-partitioned fraction pieces and asked to identify relationships between the pieces. The following excerpts are transcripts of two students’ dialogue captured on video.

Visual Transcript Student Assets

[S1 places the red whole over the blue eighths]

S1: These are eighths. Because we put eight together so they are eighths. And it is the same size as the whole.

T: You have made the connection that eight eighths is a whole. So can you make any other connections between the pieces?

The student had some understanding of the unit fraction language, such as half, fourth, third and eighth.

S2: Oh yeah – this piece. [motions to a yellow fourth]

T: OK. What is that called?

S2: It is called a fourth. So, it won’t really take as long to go together. [motions to a piece off screen] This piece would take the longest. [motions to the eighths] Well, actually this piece would take the longest. And, [picks up a fourth] this piece, all you’ve got to do is put the rest of the pieces together.

This student had made an earlier prediction about how long it would take to build the whole by using eighths and continued to think of the fraction pieces in terms of how long it would take to build the whole.

A bit later

S1: I think it is a … third.

T: OK. And why do you think that? S1: Well, it looks like less than a half but more than a quarter. [places the yellow fourth on top to compare sizes as S2 is working to build the whole with the third pieces]

This student used proportional reasoning and an understanding of benchmarks to develop an understanding of additional unit fractions.

S2: And you only have to put three together so then … [builds the whole]

T: That was fast, wasn’t it? So now you have … what would you call those?

S2: Thirds.

T: Three thirds. [points to three-thirds model]

S1 and S2: Four fourths. Eight eighths. One whole.

The students quickly adapted the language modelled to each of the other fractional representations.

Probing questions include the following:

Grade 7/8

Into a Grade 7/8 classroom, the following question was posed to students: “Out of one pan of brownies, Sandy ate 14 and Pat ate 16/ What fraction of the brownies did Sandy and Pat eat altogether? Write a number sentence to explain.” Students were provided with paper to fold to solve the task. One student folded the paper as shown. Note the space between a one-sixth fold and a one-fourth fold (when folded as shown to the right) is one-twelfth of the area of the paper, which is the common unit. The student was unable to make this connection and wrote the number sentence as 14 + 16 = 210.

Probing questions include the following:

Grade 9

A Grade 9 student was asked to solve for x in the equation 23 x + 5 = 7 + 45 x

The student completed the following solution:

Note that in the first line of the solution, the student multiplied the left side by 3 and the right side by 5.

Probing questions may include the following: