Scenario: Heidi revising percentages
Country: UK
Grade (student age): Year 8 (age 12-13)
Contributed by: Anne Thwaites, University of Cambridge, UK
Context – national, curricular, professional, other
The National Curriculum for mathematics in England includes work on fractions, decimals and percentages in Key Stage 3 (years 7-9, pupil age 11-14). Heidi was revising the four operations with fractions, before moving on to discuss some word problems related to percentages. Having completed a mathematics degree, which included an optional mathematics education element, Heidi was a graduate on a pre-service training course. The lesson took place in a school-based placement towards the end of her one-year teacher preparation.
Scenario
Heidi, a graduate student-teacher, was teaching in an open entry secondary school (pupil age 11-18) in a village in the UK. The school divides each year into mathematics groups (by ability) and Heidi was teaching one of the two parallel top groups in Year 8 (pupil age 12-13). This was a revision lesson in which she is considering fractions and percentages with the class.
After an initial activity with a fraction magic square, Heidi moved on to examples of word problems involving percentages. In working through the problems, Heidi used an interactive white board (IWB) to record the suggestions that the pupils made. The pupils all had calculators
Heidi introduced percentage revision with an example which was already on a slide on the IWB. “My rich aunt agrees to give me an increase on my allowance of 5% each month. If she used to give me £130 each month, how much will she give me now?” She asked the pupils how they are going to do the problem, rather than giving her an answer, suggesting that there are several ways and that she would show them all.
Heidi I want to know, not what the answer is. First thing I want to know is how I am going to do it. And there are a few different ways and I would like to show you all of them so that you can pick which one you like.
Nick offered x 5 which Heidi described as finding 1% and then multiplying by the “amount of percent” that you actually want. She checked if this was the final answer and a pupil says “No, you have to add it to the £130”. Another pupil, David, finds 10% and then halves it to find 5%. Heidi asked “Is there another way where I don’t have to add it in?” Laura suggested “130 x 1.05”; Heidi recorded this and then asked the class why Laura had done this calculation. Finally she asked for a calculation using percentages and was offered 130x(105/100). With four different procedures (summarised below) to solve the problem on the board, Heidi suggested that the use of decimals is preferable, as a ‘quick’ method.
(MATHEMATICAL NOTATION WILL NOT PASTE INTO DOCUMENT>>>>
Her recording of these made use of coloured pens and is shown in the figure below. It should be noted that some of the recording is not strictly accurate, particularly her use of = when she is carrying on to the next stage of a calculation.
In this cycle of our research, we were interviewing the student-teacher after the lesson. During this meeting we viewed episodes from the video of the lesson and one member of the team led a discussion with the student-teacher in the spirit of stimulated-recall. This discussion was audio recorded and included the following:
Researcher … you invite the pupils to make suggestions and recorded all the different ways, and I think ended up with four different approaches.
Heidi Yeah.
Researcher … is that a process that you actively try with this set, of getting lots of different ways?
Heidi Yeah, I think because they’re, obviously, quite a bright set I’ve sort of found that I don't want to say, ‘Right, well, this is the only way you can do something,’ because I don't believe that is the case in maths. And different people think in different ways and I do believe... I would believe more in a constructivist view of maths, and you know, I think you can discover things by yourself. And it’s not meaningful to just learn some rules. And, you know, if I tell them, ‘Do it this way,’ they might not remember it, they’ll remember the way that they understand and that they know and they’ve seen working in their head, kind of thing. …
A later part of the interview included:
Researcher ... my rich aunt, you ended up with four methods ... and you highlighted one of those, the decimal one, where they were multiplying by 1.05... and said ‘It’s a quick way.’ … I just wondered why you highlighted that?
Heidi ... I highlighted it because it was... kind of when you do it that method, like adding it onto one, it helps with the taking away from one [the second example involved percentage decrease], which isn't quite as obvious.
Researcher Oh.
Heidi … although they can do it either way, you know, they don't have to take it away from one they can find it by a percent and take it away. I think because I knew the work I’d set them for that lesson I knew it would help if they had done it... if they did it the decimal way, particularly the decrease.
Knowledge Quartet Coding Commentary
Contributed by: Anne Thwaites, University of Cambridge, UK
Knowledge Quartet Dimension: Foundation
Knowledge Quartet Code: Theoretical underpinning of pedagogy
Scenario: Heidi revising percentages
Having presented the word problem of her rich aunt to the class, Heidi takes time to say that she wants to know how the pupils have solved the problem rather than simply giving her an answer. She asks for solutions, which the class readily contribute, and she then records their solutions on the interactive whiteboard (IWB). Her use of different coloured pens to record the different solutions helps to highlight the different procedures that the pupils have suggested (as shown in the earlier figure).
Heidi is very ready to accept a range of methods to solve the problem and gives them all careful consideration by recording them on the board. Ultimately she states a preference for using decimals as an efficient (one step) procedure to solve this problem whilst she acknowledges that all the contributions lead to the correct answer.
In the interview Heidi is telling us that her beliefs about the ways that the pupils learn mathematics are based around the pupils finding a procedure, in this case, with which they feel comfortable and confident. She acknowledges that different pupils may come to this understanding via different routes and so she wants to gather differing procedures to solve the problem in order to show the pupils these differing routes. This readiness to adopt what Heidi calls a ‘more constructivist’ approach, shows us that she is thinking about the ways in which the pupils are learning as well as the content of their learning. It is interesting to note that she had a mathematics education element in her undergraduate degree program, and such matters will have been discussed in her on-going methods/didactics course.
So although Heidi highlights one particular ‘quick’ method, she is also thinking about the next section of the lesson as she describes in the interview. In the following part of the lesson there will be a different complexity in that the percentage will entail a decrease. She shows her propensity to think ahead and ensure that she has a sound foundation on which to build the next part of her lesson.
In this scenario, we see a student teacher collecting, from the pupils, a wide range of ways of solving a problem in an open and exploratory manner. In our interview with her, she readily supports this approach by referring to her beliefs about the way in which pupils learn mathematics. As she explained, “I want to take everyone’s ideas on board. When you do put something on the board they correct each other rather than me being the authority”. Her perception of this aspect of her role, as teacher, and the possibility of the pupils themselves contributing to their own learning, concurs with the ideas of various constructivist and fallibilist authors. Balacheff, for example, advised that “[the] transfer of the responsibility for truth from teacher to pupils must occur in order to allow the construction of meaning” (1990, p. 259), and identified classroom discussion as a context in which this transfer can take place. Although this type of approach is not specific to mathematics teaching and learning, from our data Heidi, particularly as a student teacher, is unusual in making such a clear attempt to allow the pupils to develop their own learning. She constantly assists this ‘letting go’ by acknowledging pupils’ suggestions, and making them available for scrutiny by writing them on the board. The successful implementation of this strategy could be construed as necessarily indicating and being underpinned by connectionist beliefs about mathematics teaching in particular (Askew et al, 1997). Her beliefs are the corner stone of the manner in which she teaches and give an example of how a student-teacher’s theoretical knowledge is manifest in her teaching.
References
Askew, M. et al (1997) Effective teachers of numeracy:report of a study carried out for the Teacher Training Agency Kings College, University of London
Balacheff, N. (1990) Towards a problématique for research on mathematics teaching. Journal for Research in Mathematics Education 21(4), 258-272.