Scenario: Eleanor discussing approaches to helping pupils solve a multiplication problem
Country: UK
Grade (student age): Primary
Contributed by: Gwen Ineson, Brunel University, UK
Context – professional
Eleanor was at the end of her one year graduate pre-service teacher training course and had completed all of her sustained school experience. She was interviewed at the end of her programme about her approaches to calculation and about her approaches to teaching calculation. The scenario below involves the multiplication of 2-digit numbers.
Scenario
Eleanor was asked what her approach would be to help pupils working on multiplication of 2-digit numbers. The specific examples are 52 x 34 and 3.4 x 4.9.
Interviewer: What strategies would you use to teach your pupils to solve the following problem:
52 multiplied by 34?
Eleanor: Again work with the units, so obviously 2 times 4 is 8, and then, yeah then the 50 times 30. 3 times, cos again yeah you can see the 3 times table in there can’t you, so there’s 3 times 5 is obviously 15, then the zero, and that’s wrong again isn’t it?
|
| 5 | 2 | |
| x | 3 | 4 |
| 8 |
Interviewer: Now I tell you what, let's leave this question and go onto this question because they are linked. This, a student has solved this problem like this, and it’s wrong, can you see why it's wrong?
[Extract from the interview schedule]
A student solved a problem in the following way:
| 3.4 x 4.9 | = (3 x 4) + (0.4 x 0.9)= 12 + 0.36= 12.36 |
How would you respond to this student?
Eleanor: Should it be 15.6?
Interviewer: Why?
Eleanor: 15.06. Because that’s going over the sort of units, is that. Instead of, cos once that gets to 10 it crosses over into the units doesn’t it? ..... So instead of being, it's 3, the answer should be 3.6 instead of 0.36.
Interviewer: Is there another problem?
Eleanor: Ah, that should be 15.06.
Interviewer: That’s actually right, 0.4 multiplied by 0.9 is 0.36
Eleanor: Oh it is. Oh ok.
Interviewer: But there is something else that they have missed which is the same thing that you have missed here. Did you ever use the grid method? Can you remember what the grid method is?
Eleanor: Oh I’ve seen them used, yeah, I haven’t used it myself.
Interviewer: Ok. The grid method uses partitioning.
Eleanor: Yes, no I did use it with them.
Interviewer: So let me start you off. Let’s go back to this one now, [the previously discussed problem] so its 52 multiplied by 34 partitioned that into tens and units. So we’ve got 50 and 2 and here we have got 30 and 4. Can you see what you need to do?
The interviewer drew the following grid:
|
50 |
2 |
|
|
30 |
||
|
4 |
Eleanor: 18? No, I’ve still got that wrong?
Interviewer: What do you think?
Eleanor: I think it's right.
Interviewer: Do you?
Eleanor: I do think it's right, or is it this, is this, that should be 1500.
Interviewer: You tell me.
Eleanor: It should be 1500, see I’m looking at it, and yeah there is no way that 30 times 50 can be, yeah, it’s the zeros. Yeah, that’s what I’m thinking back to. That’s why I’m doing that thinking no it’s got to be a smaller number.
Interviewer: Ok, so you say that you have used the grid method in schools, with years 5 and 6.
Eleanor: Yeah.
Interviewer: But you don’t feel comfortable using it yourself. Why, what methods of multiplication, let’s say where you teaching with years 5 and 6. Can you remember?
Eleanor: I was using the grid method with them, but for me I knew what I was going to be teaching so it was a case of going home and learning it to teach them. And I could do it confidently with them and explain where they were going wrong but since I’ve left it's sort of,... I’m not,... I think I’m sort of going back to the algorithms and I’m forcing myself to remember the number facts but then I can't, but then I wouldn’t record these cos I’ve only used them in the last 6 months or so, I’m not recalling them. I’d just reach for a calculator.
Knowledge Quartet Coding Commentary
Contributed by: Gwen Ineson, Brunel University, UK
Knowledge Quartet Dimension: Foundation
Knowledge Quartet Code: Concentrates on procedures (COP)
Scenario: Eleanor discussing approaches to helping pupils solve a multiplication problem
Eleanor was asked initially how she would teach her pupils approaches to solving 52 multiplied by 34. Her training had focused mainly on mental and informal approaches to calculation and she had recently taught in a vertical grouped year five and six class. Her immediate response was to write the numbers vertically and attempt the calculation using a standard but expanded algorithm, as shown below.
| 5 | 2 | |
| x | 3 | 4 |
She multiplied the units and writes 8. Then she attempted to multiply the tens (50 x 30) but got lost in knowing how to use 15 (derived from 5 x 3). She knew that her answer was incorrect but she doesn't appear to realise that she has missed stages of the algorithm.
She is then directed to another question which involves a pupil making a similar mistake. She isn't clear whether 0.4 x 0.9 is 0.36 but she is also unclear whether any other mistake has been made. She is asked about the grid method for multiplication and hazily recalls teaching it but is unable to use it for the given example. When asked about teaching it, she explained that she would prepare the night before and felt confident about teaching it, but would quickly forget the approach.
What is interesting about Eleanor's response is that she doesn't seem to have any strategy to solve either problem and makes no attempt at estimation. She may have felt uncomfortable in the interview context, especially in that the interviewer was her mathematics tutor, but her final response indicates a strong procedural approach to teaching calculation of two digit multiplication. Eleanor does not appear to have mastered the procedures for two digit multiplication either with or without decimals. Her reason for using a procedural approach would not appear to be because she felt comfortable with this but rather because she had no other strategies.
Eleanor had just completed her final nine week block of teaching experience with a class of nine to eleven year olds and would have been teaching almost full time for the last few weeks. During this time she would almost certainly have been involved in planning, teaching and assessing lessons involving calculation. Given her response, this is likely to have been procedurally driven and lacking in relational understanding.