A Companion to Research in Teacher Education

Very few teachers exhibited high-quality explanations from both mathematics
and linguistic perspectives. Task 1 showed a few teachers with this profile, but for
Tasks 2 and 3 only one teacher demonstrated competencies in both areas. The
example below is taken from a teacher’s response to Task 1.

To distribute is to share or spread things out. In the distributive property, you share or

spread a number out with other numbers and functions. For example, in 3(2 + 6), you could

add what’s in parenthesesfirst, where 2 + 6 = 8, then 3(8) = 24...or you can distribute the

3 to the 2first, where 3(2) = 6, then distribute the 3 to the 6, where 3(6) = 18. Then add the

two products, where 6 + 18 = 24. In either case, the answer is 24.

A modest number of teachers had low language and discourse quality, yet
demonstrated a sound knowledge of mathematics in their explanations. The
example below is taken from a teacher’s response to Task 2.

In adding fractions with the same denominator, determine the smallest unit fraction and

distribute this with the sum of the numerator. For example, when adding 5/8 + 3/8, the

smallest unit fraction is 1/8. 1/8 is distributed over the sum of 5 and 3, which are the

numerators of the fractions. The problem looks like this 1/8(5 + 3). You can take the sum

of 5 and 3, which is 8 and multiply it by 1/8, giving you 8/8 or 1 whole, or you can take

1/8
5 which is 5/8 and 1/8
3, which is 3/8 and since the denominators are the same,

we are adding eighths, giving us a total of 8/8 or 1 whole.

Some teachers were capable of well-structured explanations but this did not guarantee
good quality mathematics knowledge being conveyed in their explanations, although
this varied by task. Below is an example from a teacher’s response to Task 3.

I would explain that the numerators tell you how many of the denominators you have and the

denominators are like what you have. For instance 3/5 means I have three of something that is

“cut”intofifths and 1/5 means I have one of that same something cut intofifths. So if I have

three of the something and 1 of the something then I have 4 of these somethings in all. Thus if I

have 3fifths and 1fifth, then I would have 4fifths in all. I would write the explanation just like

I just did. I would also use manipulatives to show what I was talking about. The point being

that“something”which isfifths in this case does not change. It’s still the same thing.

By far the majority of explanations fell into the lowest ratings for quality of both
mathematics and language/discourse features on each task. Below is an example
from a teacher’s response to Task 3.

I would explain what the information represents i.e. the denominator (sic) represents how

many pieces the object/s are divided into and does not change. The numerator tells how

many pieces or objects there are and can be added together or subtracted from one another.

Table 47.1 Distribution of teachers by quality of mathematics and language/discourse features in
explanations

Mathematics Language/discourse

quality—high

Language/discourse
quality—low
Task 1 Task 2 Task 3 Task 1 Task 2 Task 3
Quality—high 4 1 1 8 10 8
Quality—low 19 10 6 72 56 89
NoteTask 1 (n= 103); Task 2 (n= 77); Task 3 (n= 104)

706 A.L. Bailey and M. Heritage