Relación Costo-Beneficio del Proyecto

She chose to specialise in mathematics because I quite enjoy mathematics. I think the challenges in mathematics are quite interesting. I wasn't the very best at maths, but I did

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like trying and finding out, even when I was at school... I asked for extra homework, I must be mad. So it seemed the natural choice (to specialise in). Fiona went to a private school until the age of 16. She only remembers doing School Mathematics Project (SMP) cards in lessons and loved to whizz through them. She said she enjoyed working through the textbooks doing column addition and subtraction, and had a lovely teacher so had a very good experience. She believed she has always been academically successful, as I always tried really hard in everything I did.

Fiona does not recall anyone in particular who may have influenced how she approaches mathematics teaching, but if she saw or heard of a good idea, she would incorporate this into her own style of teaching. She said I've developed my own style of teaching, like everybody does, … I've got lots of ideas from people. I'm always open to new suggestions and I've just incorporated everything into my own teaching and made it my own.

Fiona stated that she liked to challenge her pupils. Particularly now I've been working with the top ability, they enjoy the challenges and you know that they can apply themselves a bit more than perhaps the lower ability children will, they understand it a bit better. If it’s a new concept starting off with quite easy stuff and then building on that very quickly... Giving them the confidence to try and do it. When they say, I can't, I can't I can't, build on that confidence and say, have a go, it doesn't matter if you get it wrong.

When asked how the children might view their mathematics lessons she said that she hoped they'd say they're exciting. I try and cater for visual, auditory, and kinaesthetic in my lessons so hope they would pick up on that. Each of them would get something out of it, each of them would find something of that lesson very interesting. They might say some things are too hard, whereas I say it would be challenging.

Fiona enjoyed teaching shape, because you can do it very practically she said. She also enjoyed patterns, I think there's so many things you can do, making patterns with it and pictures and whatever, and addition I like teaching. Getting all the vocabulary lots of, sets of, groups of, etc. where they count how many sets, so that's so many times three blah, blah, blah.. We might stand up and do the marching and do it direct: three times dee dum dee dum... And all that...we do it in minute maths. Finding patterns in the multiplication tables, I think is quite fun. Either setting them challenges to find it themselves or, if they haven't spotted it, explaining it to them.

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To summarise, Fiona was not happy in the classroom she was in, essentially it is an expanded corridor, at one end there were two doors to other classrooms, at the other is a set of double doors to the rest of the corridor beyond leading to the hall. She found it a difficult environment in which to teach. She liked to use interactive displays that were colourful and visual, as that was her learning style she said. Resources readily available were also very important to her in supporting whole class teaching. Although there were mathematics resources available to her and her class, many were in a resource room in another part of the building, which she found difficult.

She felt restrained by the framework There's so many things packed into the framework, you never get round to teaching every single thing as you would want to teach it. You can't extend on anything too much because you've got to get onto the next area of learning. I think there's too much in it. She added I think there's too much pressure on teachers to teach everything in it, especially when it comes to the SATs. Ugh! You end up teaching things because they might be in the SATs, and that's not the way it should be done, at all. She implied she felt constrained.

6.1.3 Belief Manifestation in practice

This section of the preliminary interview provided insights into how Fiona perceived her own teaching of the subject through her pedagogical approaches in general, but in mathematics in particular and how she managed WCIT phases as well as what a typical mathematics lesson might look like.

Fiona really enjoyed talking to her class on the carpet. I like the closeness of it and having them all together, having them all listening attentively... I don't like it when they start fiddling, they have to be attentive. I especially like it when they're doing things together on the carpet in their talking partners. Although Fiona expressed a preference to closeness with her class, she emphasised that closeness was about the children being quiet and attentive. She enjoyed a starter in which she encouraged the children into the mindset of the numeracy lesson, getting their brains working straightaway, rather than having a slow introduction into the lesson and then, maybe they start to get a bit bored. She talked about how her oral- mental starter it's quick, quick, quick ...they start thinking straight away.

For Fiona the learning objectives were the most important aspect of the main phase of a lesson their understanding of what they're going to learn, the purpose of the lesson, so going

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through the learning objectives was most important, I want them to understand why they're learning what they're learning. But I want them to understand what they're going to be learning about, not in too much detail, as I don't want to bog them down. She also emphasised the importance of success criteria in the WCI phase of a lesson. I might do it straight after I've explained the learning objectives, what are you going to be able to do at the end of the lesson, and they tell me and I'll write them down. Or I might do it at the end of the lesson, or review what they've done. I think that's very important as well.

Finally Fiona said, personally, I love having them listening to me, you know, hanging on my every word, most of the time. She said in some jest.

I like the different ways that they can answer you, so I can either say, right, everybody call out at once or it'll be hands up or hands down or I'll pick children. I like it when they're in their talking partners, I'll go round and listen to what they say. If you ask somebody to come up and do something on the board we give them a cheer or clap, I like that.

She said that she encouraged this to give them confidence and to keep their self-esteem up. She said

it depends on background, home life and, how they are with their peers. So I think if you actually give that support and encouragement within the classroom, it gives them a bit of a boost. I think I try to be quite positive even when they get things wrong, I'll say, well, good try, well done, thank you for trying et cetera. But I think if you're not, if you're more strict and set in your ways, I suppose, you say, No, that's the wrong answer, next you know ...I don't think you can take that approach with young children.

She described her teaching style as friendly.

A typical lesson would be in three parts where they will do a quick oral-mental starter first. This may be with something called

minute maths where they're given a strip of sums or multiplications and I give them exactly one minute, to write down as many answers as they can, they continue that same strip of paper each week. When it's filled out, they get another one, et cetera. So it's either the minute maths or the oral-mental starter. And then... we will do a bit of a brain break. And then go on to our main activity.

A brain break refers to a whole school policy in breaking up episodes of lessons to help the attention of children. The school in which Fiona taught considered the school to be in a difficult catchment area, where there were low socio-economic family problems.

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6.1.4 Summary

To summarise this section of Fiona’s espoused beliefs and attitudes to mathematics her personal orientation can be summarised as:

 She found the subject easy at school and university

 She greatly enjoyed working through books, cards and sheets  She enjoyed the challenge mathematics offered her

 She believed the approaches in teaching are better now  She was influenced by previous teachers she has worked with.

Finally embedded in all the above were a number of pedagogical practices presented as elements of her day to day working, there included:

 Connections: Making links between the starter activity and the main part were the

only connections Fiona referred to.

 Prior Knowledge: Fiona activates prior knowledge to develop new learning

 Discussion: Talking partners were used in whole class phases to listen into the

children’s paired conversations

 Explaining: Fiona emphasised the importance of explaining the learning objective to

the children in the main part of the lesson, to ensure the children understood what they were learning. She also emphasised the importance to the success criteria, which she said she involved the children in. She also thought it important to ask children to explain to the class in whole class phases of a lesson.

 Questioning: was seen as part of developing individual’s confidence through inviting

children to answer questions, handle resources and build self-esteem. Building motivation through praise and encouragement.

 Modelling: Involve children in their learning e.g. invite children to come to the front

of the room to write or mark on the board as part of the whole class interaction between children and teacher.

 Resources: Fiona believed that resources were essential as children are motivated by

handling resources.

 Fun: Fiona believed she emphasised fun lessons to motivate learning of the subject

 Flexibility: She believed that learning styles were very important in her approach to

planning teaching and learning.

 Brain-breaks to aid concentration: Fiona believed it important to have a brain break

during lessons to aid concentration of the children. She believed that a very important aspect of her role was to motivate children and build confidence through praise and development of self-esteem.

111 6.2 Overview of Practice

This second section presents a summary of the three lessons observed and then later discussed through stimulated recall interviews (SRI) to allow reflection. The presentation of this section has been separated into three categories:

 mathematical Intention;

 pedagogical approaches

 classroom norms (behavioural approaches).

Each category presents the data that characterise the enacted practice observed in the three lessons. Each item has been supported by examples of events consistently observed or comments consistently made or emphasised by Fiona in the SRI providing evidence of the way in which she conceptualised her practice. Comments made by Fiona are represented here in italics.

6.2.1 Mathematical Intent

Prior knowledge was activated in every lesson observed through Fiona’s explicit statements

and her questioning in response to children’s comments, for example she would ask the class if they remembered how they had added or subtracted in previous lessons. In her post lesson interview we discussed a question one child asked her following her explanation that they were going to count in twos. One child shouted out What does that mean? She said I'm sure we've done counting in twos before but not like that. She thought they were a little confused as she was using money (2p coins posted into a money box). She added the focus was counting in twos not money, although she did not make that explicit to the children in the lesson. She emphasised in her interview that ...because it was something different to do, something a bit more challenging than just sitting there reciting in twos. Just to get them a bit more stimulated. Fiona was very keen not to give children mundane ‘boring’ activities which will be discussed in the next section - pedagogical approaches, in more depth.

Mathematical vocabulary: Fiona consistently emphasised key mathematical vocabulary

throughout her lessons. She would repeat the key words over and over to the children and ask them to say it after her, sometimes writing them on the board. Following the framework was the rationale she gave for this action, she said I want them to begin to recognise the words when they're written down, because in the framework one of the things is to begin to write down.

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She was also seen to use games to emphasise children’s use of vocabulary, or particular phrases, drawn from the framework. When discussing one activity she said

I wanted it to be a different activity coming away from the number lines, because they had been working so hard at that, and I chose the box (a covered shoe box), because I wanted them to think about the vocabulary that's involved, using subtraction as well using less than.

In one lesson Fiona explained that she always read out the learning objective as they were presented on the class board just as they are in the strategy framework. She said the learning objective is from the framework so I wanted to make sure that the children understood what it meant, in their vocabulary, e.g. to partition a two-digit number. Fiona often talked to the children about mathematical vocabulary and corrected them when they got it wrong e.g. repeating the words correctly as in one lesson ‘seventeen, not seventy’ emphasising the endings, but was not seen to relate the numbers to the system through the use of a number line or grid etc.

Connections: Although Fiona made relational links between concepts within mathematics,

the emphasis was on the resource and not the mathematics. For example: between IWB place value cards and partitioning with the use of cubes to illustrate the concept of partitioning two digit numbers. She said

I've used those two together before and it's been successful. I used the Place Value cards that they hold which are better. This was good because you could see how the numbers began in their tens and units and you could put them together and partition them again, like the children needed to know. What I would have liked to use are the smaller Place Value cards at their tables, but unfortunately we didn't have the resources for that. So the next best thing was cubes... and to try to relate the partitioning and making towers of ten to relate to the numbers.

I asked if she felt restricted by the fact that she could not use the resources she would have preferred to use, but she had not thought about there being an issue if the resources were not available to her.

To summarise these points, it was apparent that Fiona had little to say about the children’s mathematics learning and understanding, she emphasised a methodical yet procedural or Instrumental (Skemp, 1976) approach to teaching mathematics through each of the above features. For example, Fiona appeared to reactivate prior knowledge by asking questions about previous mathematics lessons, however the prior knowledge was not seen to be related to different aspects of mathematics or indeed other areas of the curriculum and

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children’s learning. Instead it related to the context or the resource used. Key vocabulary were explained to the children in great detail through definition and fact, little or no explanation or relationship made for the children to make explicit connections with other words or learning. Fiona provided detailed mathematical learning objectives as already alluded, however, the explanations were about ‘what’ the children are learning, not ‘why’, other than the children need to know this.

Fiona was filmed in two different classes of a year one and a year two, the mathematical emphasis appeared similar in approach and presentation. She explained that she believed both the classes were well below the national average and was therefore a difficult task. She talked about ability groupings but little emphasis was recorded in whole class phases of a lesson that supported that idea, most discussion was focused on the lowest expectations. The differentiation appeared solely to be emphasised in the seatwork that followed WCI, e.g. seen through the numbers the children were working with e.g. 0-5, 0-10, 0-20 or 20-100.

6.2.2 Pedagogical Approaches

Fiona emphasised her pedagogical approaches in her initial interview, however, many more were revealed through the observation of her lessons. This section is again presented through themes drawn from the individual case data coding (example of which can be found in Appendices: 6.1 which presents the first level of analysing lesson content against SRIs, and 6.2 which presents the second level of analysis of lesson under categorisation headings).

Resources, visual aids and practical equipment.

Fiona was observed to use many practical pieces of equipment in her teaching, for example she used two pence coins dropping into a metal money box and asked the children to ‘visualise’ the counting of the coins (counting in twos). Shopping items were used to identify 3-D shapes, she said

well I didn't want to use the same old boring things out the drawer that we always use that don't have any relevance to real life really ... so I raided my cupboard to find some different shapes. Everyday items that they would see, for them to realise that shapes exist in real life not just in school or out the drawer.

Not all items were explicitly typical, for example a peanut butter jar was introduced as a cylinder.

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She was regularly seen to use a covered shoe box (similar to figure 6.1 below) which contained a set of cards, either with questions such as 7+4=? written on them, or a number between 0-20. The children appeared to be familiar with this box of cards which implied this

was a regular resource used by Fiona. When discussing its use she said I chose the box, because it's exciting picking something out of a box, because they don't know what they're going to get. She added that she believed the use of the box involved all the children so that the whole class is thinking about that number not just that one child, also to help that child if

they did get stuck, to give them some support from their peers, is just a nice practical way of doing it.

Figure 6.1: Covered shoe box

Linking cubes (like the cubes in figure 6.2 below)were used in one lesson to make towers in support of children’s understanding of partitioning. She explained that she used them to