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Maths (Teaching Primary Mathematics, Turner (Differentiated Teaching (Pg.…

- - - - Behaviourist
        Didactic
        Characteristics:
        learning by transmitting knowledge, telling information, giving instructions to the learner
        teacher controlled
      - Cognitivist
        Use of orange of styles that stimulate more senses and forms of communication than hearing, i.e. | speaking, using visual aids, demonstrating, etc, Children assess their own performance against criteria
      - Constructivist
        Child-centred
        Characteristic
        investigative approach to learning reflecting on action to make generalisations teacher acting as facilitator teacher-and child-controlled guided discovery
  - - - Differentiated teaching is not an approach but is based on the ideas of how children learn discussed in Chapters 3 and 4. The child is not seen as a receiver of teacher input from which output is expected; rather the child is seen as an active participant in the learning process, engaging with teacher input by finding ways to understand what the teacher offers whether it be facts or experiences.
      - Maintaining motivation is a key element of effective teaching and depends on knowledge of the child as well as the subject content, Equally, no one approach can do this for all children so teachers must tailor their approaches to meeting the needs of the majority of children while providing different experiences for others.
  - - - Shulman (1986) devised the term ^ Pedagogical Content Knowledge (PCK) to describe the interrelationship between content and pedagogy. He carried out a longitudinal study of secondary teachers in their training and beginning of their teaching career. He believed there to be a symbiotic relationship between what he called'subject matter knowledge' and 'PCK' i.e. the process of developing lea to stronger forms of the subject matter knowledge.
- - - - participating in effective group work, not only in terms of insights gained from the contributions of others but also through having an opportunity themselves. The second is the crucial role of teachers in enabling successful collaborative talk and using the insights gained from it in assessing their students' progress and planning for future learning.
      - work of Vygotsky (1986) • who argued that children learn through their interactions with (more capable) peers and adults.
        proposed that engagement in intermental activity (social interaction) fosters the development of intramental (individual) cognitive abilities
  - - - The representations and contexts we use in classrooms to model the mathematics such as fractions of cakes number lines hundred squares context of money for decimals-ideally bring meaning by providing connections with what is already known intuitively by the learner and the mathematics under consideration. But such representations have their limitations: oaf model or context will not represent all of the mathematics and the learner will need to experience several models and be able to move flexibly between them to successfully build the mathematics (see Delaney, Chapter 5, and Barmby, Harries and Higgins, Chapter 3).
      - We use the word to refer to the way mathematics is connected with a 'real' everyday world - the everyday world being then represented t?y mathematics. 'One can say perhaps when a child has a "modelling error" the child has their own "model" of the situation in conflict with the "mathematical model" expected in the academic context of school' (Ryan and Williams 2007:16).
  - - - Much classroom discourse consists of what Bloom (1956) terms low-level questions for which the teacher knows the answer (for example, asking children to recall facts and procedures: what is 8 times 6?). Generally- increasing the proportion of higher-level questions is associated with increases in children's understanding (Burton et al. 1986). Higher-order questions require children to think (for example, 'How would you work out 8 times 6 if you didn't know it?')- Such questioning is often focused less on what children already know (although this may be important) and more on what they need to do in order to learn more. But the situation is not straightforward. Whether a question is actually challenging depends on what the children already know. So, a high-level question for one class may be a low-level question for another. come up with 'good' questions a teacher needs to know the children. But higher level questions are also 'harder'. Children need to think and thinking takes time.
      - But, while wait time is very powerful increased wait time is not an effective strategy for lower-order more straightforward questions (for example, recall of number facts). Somewhat paradoxically increasing wait time than about 5 seconds can actually decrease the quality of classroom talk (Tobin 1986). In fact if children cannot respond within 3 seconds, this may indicate that they need time to talk or work on the problem in pairs or small groups.
      - Talking is central to teaching mathematics formatively. But talking is not in itself enough. Talking allows children to express what they know. But they then need feedback on what to do next. Providing feedback in mathematics can be tricky. Part of the power of mathematics is that ideas can be expressed very concisely. Yet this strength can make mathematics difficult to teach and learn. Through exploring and 'unpacking' mathematics children can begin to see for themselves what they know and how well they know it. By listening to and interacting with pupils, a teacher can provide feedback that suggests ways in which pupils can improve their learning.
        If we want to find out what children understand in mathematics rather just what they can recite then we need to challenge children with activities that encourage them to think and talk about their ideas. This may involve presenting children with the unexpected: an 'obvious' answer that is in some inadequate, a problem that does not have just one correct answer or a teacher defending the 'wrong' answer.
    - - teacher working with a class of up to 30 children will clearly not be able to spend lengthy periods of time working one to one assessing all the children in depth, and this level of assessment is probably unnecessary for the majority of children. What teachers can do however, is encourage children to talk to them about mathematics, and to them, not in order to identify 'wrong' answers and put them right, but to enter more fully into the children's minds so that they can build on A second issue for teachers is to consider
      - A second issue for teachers is to consider the extent to which children are perceived as successful at mathematics may be learning procedures without developing matching mathematical understanding. This can lead to children experiencing difficulties in mathematics later as each procedure has to be memorised individually because they cannot make conceptual links what they have learned already.
- - - - differentiation by resources;
        Many children find the move from concrete representations of mathematical ideas to more symbolic or abstract representations difficult (Hughes, 1986). Providing children with concrete materials to support their mathematical thinking is common practice. As children's competence and understanding develop, they become better able to use and manipulate abstract symbols. One way of differentiating mathematical tasks is to provide more resources for children who are not yet ready to make the move to more abstract ways of thinking. This form of differentiation enables all the children class to work on the same mathematics, but with different resources.
      - differentiation by peer support;
      - differentiation by adult support;
        There are several ways of differentiating work by the level of support. You may choose co ask your TA if you are lucky enough to have one, to work with a particular group or roups. Differentiation is about making the mathematics challenging and appropriate for all the children in the class. The TA could equally work with any group of children in the class. You may choose to work with a particular group of children to provide additional support or challenge.
      - differentiation by outcome;
        is considerable evidence (Boaler, 2010) that children derive a great benefit from working together in 'mixed ability' groups. It is possible in this situation that the children could all be given the same task or activity and that the differentiation comes through the outcomes that the children produce. Planning mathematics lessons in this way requires a different kind of mathematical task, one that can be characterised by the phrase 'low threshold high ceiling'. Such tasks are simple enough for all the children to be able to make a start and to contribute something, but are not limiting so that some children car them mathematically more demanding. Planning such tasks is more demanding than giving children closed tasks where there are specific answers, but arguably leads to much richer mathematical learning. Planning and teaching in this way requires a shift focus away from answers to the children's mathematical thinking and reasoning
      - differentiation by choice.
        Consider the possibility of giving the children a choice about the nature of the resources chat they might use to tackle a particular mathematical task. Clearly you should retain a degree of control and you may well want to encourage certain children (who are possibly a little over-optimistic) to use resources and others (who are possibly less confident or who just enjoy an easy life) to try tasks without the help of particular resources. Nobody knows the class as well as you do, so you are in a good position to make chose decisions. More importantly, by giving an element of choice to the children, you are conveying important messages about autonomy and independence and about taking the children seriously as learners who have valid opinions and views about their own learning.
  - - - Dyscalculia affects 'arithmetic skills'. We need to remember that arithmetic is just one small part of the larger subject we call mathematics. People with dyscalculia do not necessarily have any problem with other mathematical topics or with mathematical concepts. It is perfectly possible for someone with dyscalculia to study science subjects to degree level, despite a continuing difficulty with basic arithmetic.
      - The definition includes both number facts and procedures. An example of number facts might be all the times tables facts up to 10 x 10. An example of procedures might be all the sequential steps needed to work through the standard algorithm for division. The salient point is that both aspects, number facts and mathematical Drocedures can be affec
      - The final sentence in the definition implies that teachers must aim higher than just teaching children to produce a correct answer or use a correct method'. Most maths educationalists strongly agree that teaching children to find answers mechanically is simply not good enough. A child who just goes through a series of prescribed memorised steps has only learned how to follow a recipe not how to really understand, or undertake, the maths.