Scenario: Bella discussing connections
Country: UK
Grade (student age): Trainee teacher
Contributed by: Gwen Ineson, Brunel University, UK
Context –professional
Bella is at the end of her one year post-graduate teacher training course. She has completed nineteen weeks of sustained school experience and her most recent experience was in a year 1 class of five and six year olds. She was engaging in a discussion about her mathematical subject knowledge for teaching and how she would respond to pupils encountering difficulty with various numerical problems. Questions included subtraction of a two digit number from a three digit number (234 – 48), multiplication of two digit numbers (52 x 34), multiplication of decimals (3.4 x 4.9) and division of fractions (1 ¾ ÷ ½) . The transcript included below is an extract from Bella’s response to the division of fractions question.
Scenario
Researcher: Division of fractions is often a little confusing for pupils and teachers. People have different approaches to solving problems involving division with fractions. How would you solve this one? (1 ¾ ÷ ½).
Bella: I know you need to find a number that… I’m going to say that they are all divisible by, so that’s 12.
Researcher: OK, you can use 12.
Bella: You can use…. cos the 3…., cos originally I was going to use 8 but then obviously 3 doesn’t go into 8. So, so I would change these 2.
Researcher: OK.
Bella: You need to add that into it, that needs to be added into that fraction. So that’s, is it another lot of 12? [Indicating the 1] So its 3 over, no that becomes a top heavy fraction so it’s, is it 15? Yeah its 15 over 12, divided by 1ower…
Researcher: Let’s go back to this as it is now [Pointing to the original question]. What is the question asking you? What’s another way of saying that?
Bella: So you’ve got 1 and ¾ divided by half, right so that is, 25, yes 1.25.
Researcher: Tell me what you are doing.
Bella: I’m thinking of 1 and then I’m thinking of it as a decimal and thinking that 0.25 goes into 1, 4 times, so that’s the 4 quarters.
Researcher: Keep going
Bella: So that’s 4 quarters but I only need 3 of them, so 3 times 25 is 75, so it’s 1.75 and then just halving that.
Researcher: Well you started well. Go back to what you were talking about, you said that there was those many of those in those, [pointing to the 4 lots of 0.25 indicated above] what were you talking about then?
Bella: ¾, cos I was turning them into decimals, cos I was going from fractions into decimals.
Researcher: So, ok let’s change that so … what’s that as a decimal, 1 and ¾?
Bella: 1 and ¾, 1.75.
Researcher: Yeah, that’s what you got 1.75.
Bella: Yeah…See this is the problem that I have got, I’ve got all these little facts and they are all over the place. So that, so somehow I know that obviously three quarters is 0.75 and then you times it by 10 and it’s 75, I’ve got all these useless, they are not useless, but, well, they are useless unless I can apply them.....See that’s the thing, I’ve got all these jumbled up facts and I know all these silly little things, that that needs to be that but I don’t know why, there is no sense to it.
Knowledge Quartet Coding Commentary
Contributed by: Gwen Ineson, Brunel University, UK
Knowledge Quartet Dimension: Foundation
Knowledge Quartet Code: Overt display of subject knowledge (OSK)
Scenario: Bella discussing connections
During the interview Bella recognised that her mathematical subject knowledge was insecure in some areas. She explains that there are bits of the subject that she understands, but she seems to recognise the need to understand how each part is related to another. Her approach to solving division by ¾ was to convert to decimals, hence her first comment above about ‘little facts that are all over the place’. She had panicked about having to solve a division of fractions problem and had attempted to remember what to do. She recognised that without being able to remember the rule, she was unable to attempt the calculation, or indeed estimate the solution. However, having converted the fractions to decimals, she was still unable to attempt the problem.
Bella then suggests that she knows how to multiply decimals by ten (although she makes a mistake) but the implication in what she says is that she understands these procedurally and doesn’t understand how the procedures ‘work’. This indicates a recognition that different elements of mathematics are connected, but she doesn’t know how and is therefore unable to make use of this in her work. Without observing Bella teaching it is impossible to know the impact of this understanding on her teaching, but her desperation suggests that she would at least attempt to highlight connections between concepts with her pupils.