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# Ei505 maths 1

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• Discuss BrunerBruner (1966) suggests an important model for depicting levels or modes of representation. One can experience and subsequently think about a particular idea or concept on three different levels: enactive, iconic and symbolic. These ideas have been extended by Lesh (1979) and are discussed in the next section. At the Bruner enactive level, learning involves hands-on or direct experience. The strength of enactive learning is its sense of immediacy. The mode of learning Bruner terms iconic is based on the use of the visual medium: films, pictures, diagrams, and the like. Symbolic learning is that stage in which one uses abstract symbols to represent reality. (Post article)For example, consider the operation &quot;two plus three.&quot; From the child&apos;s perspective this idea is experienced enactively if the child joins a set of two objects with a set of three objects and determines that there are five objects altogether. This same notion is experienced iconically if the child views a series of pictures. The first might have two objects (birds, children), which are joined with a set of three objects in a second picture. The third picture might show that here are five altogether. Note that at the iconic level the determination of the result, five, is actually made by the developer of the diagram or photo, not by the child. The relationship is symbolically encountered when the child writes 2 + 3 = 5. Bruner contends that all three types of interpretations or modes are important and that there is a common sense order implied by three levels because each requires familiarity with the earlier modes of representation.Implicit in his and later work (Lesh, 1979) is the fact that these modes should be interactive in nature, the child freely moving from one mode to another. For example, given the equation 2 + 3 = 5, the child could be asked to draw a picture of this situation. This would in effect be a translation from the symbolic (2 + 3 = 5) to the iconic mode (pictures).Your representation of 5. Was it enactive, iconic or symbolic – or a combination&gt; How do the resources fit in to this? Which use symbolic only? Which use enactive? A combination of more than one mode? How could you represent in that way. Think about this when you look at the next problems.Bruner and Skemp and two names that we will refer to in most sessions and are key to our beliefs around primary mathematics teaching.
• How do they compare in terms of progression? Expectations
• Give example of cross curricular topic work where big numbers involved and we are interested in the thinking not the calculating
• First bulletThis ensures number understanding is secure – quantity value and relative size/position emphasised in mental calculation. These are important precursors for understanding written methodsMental calc strategies do not spontaneously arise in all childrenNew calc methods shoudl be shown alongside a previously well understood method to support transition from one to anotherJust because we know how to add in columns, doesnt mean it is the only one we use. See new NC, emphasises mental calc throughout KS2Problem solving gives reason for doing this work, and need to find ways that skills practice involves thinking as well as doing to avoid tedium (see next slide and then next group activity)Place value a key issue in compact methods and will always need to consider place value when errors arise.
• https://studentcentral.brighton.ac.uk/webapps/portal/frameset.jsp?tab_tab_group_id=_314_1&amp;url=%2Fwebapps%2Fblackboard%2Fexecute%2Flauncher%3Ftype%3DCourse%26id%3D_25291_1%26url%3Dhttps://studentcentral.brighton.ac.uk/webapps/portal/frameset.jsp?tab_tab_group_id=_314_1&amp;url=%2Fwebapps%2Fblackboard%2Fexecute%2Flauncher%3Ftype%3DCourse%26id%3D_25291_1%26url%3Dhttps://studentcentral.brighton.ac.uk/webapps/portal/frameset.jsp?tab_tab_group_id=_314_1&amp;url=%2Fwebapps%2Fblackboard%2Fexecute%2Flauncher%3Ftype%3DCourse%26id%3D_25291_1%26url%3D
• Reveal on grid multiplication18 x 23What about decimals?
• Multiplication
• DivisionModel this as an approach to differentiationWhen they have engaged with the task – where is the differentiation?Where is the U&amp;A:Making reasoned choices about method, numbers, initial chunk to be subtracted
• Addition / subtraction with decimals – includes estimationSee separate menu on Word doc
• ### Ei505 maths 1

1. 1. EI505 Computing and Contemporary Developments NC 2014 Update Primary Mathematics 1 11th Feb 2014 Diana Brightling
2. 2. Session structure • Revisiting EP104 and reflections on teaching maths on 1st placement • Key features of primary mathematics in the new national curriculum • Progression in arithmetic (NC 2014) • Implications for EI505 Assignment
3. 3. Looking back at Year 1………… • Key learning from EM402 • Key learning from practical experience of teaching maths on placement 1
4. 4. Relational and Instrumental Understanding • Relational understanding is….”what I have always meant by understanding: knowing both what to do and why.” • “Instrumental understanding I would have until recently not have regarded as understanding at all. It is what I have in the past described as ‘rules without reasons’ without realising that for many pupils and their teachers the possession of such a rule, and the ability to use it, was what they meant by understanding.” Skemp, R. (1976) “Relational understanding and Instrumental understanding.” Mathematics Teaching, 77, pp 20-26.
5. 5. Bruner’s modes of representational thought enactive The child needs experience at all three levels iconic symbolic Text provides experiences at only the two most sophisticated levels Delaney, K, (2001), ‘Teaching Mathematics Resourcefully’, in Gates, P (Ed), Issues in Mathematics Teaching, London: Routledge Falmer (available as an e-book)
6. 6. Where are we now? • The new national curriculum for Year 3, Year 4 and Year 5 will come into force from September 2014. (Current NC has been disapplied for these cohorts) • The new national curriculum for Year 6 will come into force from September 2015.
7. 7. The ‘old’ National Curriculum Attainment targets: • Ma1: Using and applying mathematics • Ma 2: Number • Ma 3: Shape, space and measures • Ma 4: Handling data The ‘new’ National Curriculum 2014 Number – – – – – and place value addition and subtraction multiplication and division fractions, decimals (Y4+) and percentages (Y5+) Ratio and proportion (Y6+) Algebra (Y6+) Measurement Geometry - properties of shape - position and direction Statistics
8. 8. Some key difference between mathematics in the old and the new NC • More detailed – and now set out in Year groups. A ‘mastery’ curriculum. NC ‘levels’ have gone • More ambitious expectations, especially for number. • Greater emphasis on arithmetic – especially formal written methods • Almost no mention of problem solving, reasoning or communicating in the Programmes of Study – although these elements are there in the introduction.
9. 9. The new National Curriculum: Aims The national curriculum for mathematics aims to ensure that all pupils: • become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately • reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language • can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.
10. 10. Catering for the needs of all pupils…. The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.
11. 11. Excerpt from NC 2014 mathematics programme of study (p100) Information and communication technology (ICT) Calculators should not be used as a substitute for good written and mental arithmetic. They should therefore only be introduced near the end of key stage 2 to support pupils’ conceptual understanding and exploration of more complex number problems, if written and mental arithmetic are secure. In both primary and secondary schools, teachers should use their judgement about when ICT tools should be used. What does this mean? Are we allowed to use calculators in primary schools and if so when and what for?
12. 12. What is the progression through the four operations, from mental strategies to compact written methods? Sort the cards, putting the images in order of progression. Consider: • Is any stage missing? • How each method relates to the National Curriculum 2014 calculation strand • Where are the tricky transition points? Why? • Which key resources support understanding by clearly modelling the method?
13. 13. Key principles in supporting the development of written calculation methods • Mental calculation confidence should be established before written methods are introduced • Mental calculation strategies need to be specifically taught • We need to carefully structure progression into written methods to ensure each new method builds on understanding • Children need to be encouraged to make decisions about which method to use and when • Opportunities to apply and problem solve with calculation skills and strategies should run alongside practice of them • We need to ensure that we use resources to support understanding of how methods represent number quantities
14. 14. New ‘Mathematics Education’ Area of studentcentral (within ‘My School’ area) https://studentcentral.brighton.ac.uk/webapps/portal/frameset.jsp?tab_tab_group_id=_314_1&url=%2Fwebapps%2Fblackboar d%2Fexecute%2Flauncher%3Ftype%3DCourse%26id%3D_25291_1%26url%3D • A range of resources to support teaching and learning • E.g. Screencasts • NS ‘Strand’ documents
15. 15. Problem solving with the grid method x 10 3 200 24 Which two numbers have been multiplied together in each grid. How do you know? Multiplication grid ITP
16. 16. TARGET >300 x >300 Shuffle some digit cards and make a stack. Turn over one card at a time and decide together where to put it. Will your product be more than 300? Five points if it is. How do you know? When do you know? Use grid method to calculate the answer. Play against or with a friend
17. 17. Division Practice 245 642 563 126 246 487 623 399 280 450 266 511 188 216 160 4 8 6 7 5 3 • Look at the numbers in the yellow cloud and the numbers in the blue cloud. • Choose a number from each cloud and create a division calculation • Solve the calculation by chunking on a number line or using written chunking • See next slide for options
18. 18. Division practice options Option 1: • What remainder do you get when you have divided your numbers? • Put a cross or counter on the grid to match this remainder. Can you get three crosses in a row? 1 4 3 2 3 1 5 6 2 3 2 1 Option 2: Work with a partner to solve some division calculations using the cloud numbers What do you notice? How can you make your chunks efficient? (use the smallest number of chunks) What makes the answer smaller or larger? Can you predict if you will have a remainder and how much this remainder will be?
19. 19. Menu You have won a prize in a competition – a free meal at your favourite pizza restaurant! You want to gain the most possible from your £20 prize but cannot spend more than this amount. Which choices would you make if you choose one each from the following: • Starter • Main course • Desert • Drink?
20. 20. Considerations for your assignment?
21. 21. Follow up from this session: • Revisit and update your primary maths tracker – especially the action plan (and upload this to your eportfolio, tagging it to TS3) • Familiarise yourself with the resources in the Mathematics Education area of studentcentral • Familiarise yourself with mathematics programme of study for KS2 in the 2014 National Curriculum and English specialists – consider possible content choices for your assignment.