# Video Study 1 – Video Study 2

## Video Study 1 – “Thinking about Fractions: moving beyond one half and two quarters”

In this video clip, two Grade 4 students are thinking about the fraction 2/5 (two fifths). The two students have explored simple fractions such as one half;

three thirds; two fourths or two quarters previously in Grade 3, but have not encountered this ‘unfamiliar’ fraction. In this preliminary task, the teacher planning team wanted to observe how students thought about unfamiliar fractions without prior direct instruction. The teachers considered this math task to be a diagnostic assessment opportunity that would reveal the misconceptions about fractions that students were working with in the fall of Grade 4 (age 9) in Ontario, Canada.

The task students were assigned was set out as follows:

- Work with a partner.
- Choose one fraction – either 2/5 or 1/10
- Represent the fraction in several ways using manipulatives/materials available on the table and using markers and poster paper.

*Now let’s watch the video!*

### Transcript

St1: Okay so we, to figure out two fifths, we made rectangle with 3 lines vertical and 2 lines horizontal.

R: How many lines vertical?

St1: Um Five.

R: Okay why did you use 5 and 2 again?

St1: Because in two fifths, there’s a 2 and 5.

R: Okay

St1: So we did 5 two times and got ten.

R: Oh, so you found five and coloured it in?

St1: Yes

R: and then you found another 5 and coloured it in.

St2: and then we did uh two groups

St 1: The same

St 2: yea

St 2: But instead we coloured in two, five times. Like two squares, five times.

R: And what does that show?

St 1: That shows 10.

R: So you’ve got ten squares coloured in your…?

T: Ah, one, two, three, four five, six, seven, eight, nine, ten. Ah!

St 1: Cause we did, on this one we did five, two times I think. One two three four five.

T: So Eva does this have something to do with your last statement?

St 2: Yes.

T: So explain the last statement then.

St 2: Okay um I thought that two fifths might equal one tenth so we tried it with a chart. Because it does say two fifths.

St 1: So we are still trying to figure out if it is like multiplication, addition or division.

St 2: I think it is like addition because we have like, two fives.

St 2: No but two plus five equals seven, and two *times* five equals ten.

R: So can I just ask – would it be fair to say that this one is showing two fives and this one is showing one ten?

St 1: I guess.

R: Is it?

St 1: Yah.

### Video Guide Questions

- What assets are the students demonstrating in terms of their understanding of number sense?
- What do the students seem to understand about fractions?
- What do the students seem to be lacking in understanding about fractions?
- How are fractions like division?
- Do these students understand what two fifths means?
- What would you do next, as the teacher, to help these students with their understanding of two fifths (and of fractions)?

## Video Study 2 – “Two grade 5 students thinking about the fraction four tenths”

It is important to know that no formal fractions instruction has occurred prior to this video clip. This is an exploratory task where students are asked to share their thinking so that teachers and researchers can learn more about how children are actually reasoning when they are considering fractions. This helps us as educators to better reveal the challenges students are having rather than masking fragile or conflicting conceptions. You will notice in this clip that the researcher who is talking with the students does not jump in to try to ‘correct’ thinking because the goal of this activity was to see how students are thinking about fractions, not teaching the meaning of 4/10ths.

In this video excerpt a male and female student are thinking about and playing with the fraction four-tenths in different ways:

1.as an area model (colour in four of ten squares)

2.as a set model (four of ten pieces of pizza are eaten – or ‘taken away’)

3.as a physical representation of four red on top of 10 other coloured tiles (this may be connected to the language of “four over ten” which gives students the impression that there are two whole numbers oriented so that one number is on top of or ‘over’ the other number)

4.as a ratio (4 red squares: 10 other coloured squares).

Perhaps in watching the video, you will see some other ways the two students are thinking about fractions that are not identified here.

When you are watching the video, consider the following three questions:

- What do you think are the assets of these two students as learners?
- What do they understand about fractions?
- What do they not seem to be struggling with in their understand about fractions?

*Now let’s watch the video!*

### Transcript

St 1: No its…I already made ten, I already made ten. Two, four, six, eight, ten.

St 2: And now we need four… for the top. So we know that we’ve covered those ones in.

R: Okay, so tell me what you were thinking.

St 1: So we made ten squares

St 2: And then we put four over top of the 10 so we know which one’s we covered in.

St 1: So its like… (laughter) Okay. And then you put one…two.

R: So you’ve got like a base of ten?

St 1: Yes.

R: And then what did you do?

St 2: And then we put, we took out four of it
St 1: We took four squares and then put them on top of…

St 2: So we know that those are the ones that we’d take away.

R: Okay, so how does that show, you’re working on four tenths right?

St 2: Yeah.

R: How does this show four tenths?

St 2: Cause there’s ten, there’s ten squares and then we put four over top of it for it to… take away four from it.

R: To take away from? So you’re subtracting these?

St 1: I think so.

R: What does these four red mean?

St 1: These four red means like, if these were all yellow or the same…

St 2: And if you had pizza, you ate four pieces out of then ten the ten pieces of pizza

St 1: Like we did here, we coloured them in so like colouring them in.

R: So this is sort of like colouring in?

St 2: Yeah.

R: I see, so you’re showing ten squares and four of them are coloured in.

St 2: Yeah.

R: … kind of thing.

St 1: Just like Colby’s only a little tiny different. Um, there’s the four that’s coloured in, and then here’s the ten that’s not coloured in, that’s… if you take away four it’s gonna be six.

### Video Guide Questions

- What do you think are the assets of these two students as learners?
- What do they understand about fractions?
- What do they not seem to be struggling with in their understand about fractions?
- What role does colour play in helping students to illustrate their thinking?
- How does subtraction play into their thinking about fractions?
- Can we think about the fraction 4/10ths as a ratio?