Mathematics: Calculation Oakham C of E Primary School 1.The new curriculum 2.Calculation 3.How you can help The new maths curriculum Fluency Reasoning Problem Solving 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 have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems Efficiency Accuracy Flexibility Procedural fluency Conceptual fluency How would YOU solve these? The National Curriculum for mathematics aims to ensure that all pupils: Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language. The ability to reason mathematically is THE most important factor in a pupils success in mathematics. Always? Sometimes? Never? 1.If you multiply an even number by 5, the answer is a multiple of The best way to sum a group of 2 digit numbers is to use a written method. 3.You can always half a number exactly if it has a two in it. 4.An even number divided by an even number equals an even number. 5.Multiplying gives a bigger answer than the number you started with. The National Curriculum for mathematics aims to ensure that all pupils: 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. Mastery Curriculum Just getting the right answer in maths class isnt enough if students dont know why the answer is the right one. National Curriculum 2014 Mastery Curriculum Expectation that ALL children are capable of achieving high standards. Expectation that children will move through the curriculum at broadly the same pace. Children assessed regularly to enable intervention to be targeted. Rapid grasp of concepts will be challenged to develop a deeper understanding rather than moving to the next years objectives. Those not fluent enough will consolidate understanding before moving on. Basic Mastery Deep Assessing depth of understanding Work out: 8.4 x x x 5 0.67 x x x 93 Mastery (being able to meet expectations) Year 6 Assessing depth of understanding Which of these calculations would you prefer to work out? Explain why. 35 x x 0.7 Or 3.5 x x 7 Depth (exceeding expectations) Year 6 Calculation: Beyond Counting Understanding of place value Calculation: Beyond Counting Understanding of place value Our number system is Base 10. H T 1 52 Calculation: Beyond Counting Have a go counting with Base 5 ! Try writing 1 20! Tw F Calculation: Beyond Counting What number is this? Tw F 1 32 Calculation: Beyond Counting Understanding of place value 4 operations: +- x Calculation: Beyond Counting Perceptual Variation: Seeing in different ways. Supports: development of strong visual models finding models that work for each individual exploration of new ideas / concepts expression of ideas in convincing others Calculation: Beyond Counting Perceptual Variation: Seeing in different ways. MYTHS! Good mathematicians: dont use equipment to learn do it all in their heads are the fastest to the right answer are born that way! Maths Super Powers Conjecturing and Convincing Simplify and Generalise Imagining and Representing Classifying and Organising Professor John Mason (2005) Calculation: Beyond Counting Perceptual Variation: Seeing in different ways x 4 Calculation: Beyond Counting Perceptual Variation: Seeing in different ways. Counters Straws Egg boxes Hoops Number lines Place Value Counters Deinnes Jottings/drawings Written expression Expanded (partitioning) Column method +/- Grid method Long multiplication Bus-stop Mental strategies Written strategies Calculation: Beyond Counting Division: sharing vs grouping Sharing 8 4 = 2 Calculation: Beyond Counting Division: sharing vs grouping SharingGrouping 4 = 2 Calculation: Beyond Counting Division: from sharing to grouping SharingGrouping 4 = 2 commutability inverse of X Calculation: Beyond Counting Multiplication: from number line to formal 3 x 4 Calculation: Beyond Counting Multiplication: from number line to formal x 4 Calculation: Beyond Counting Multiplication: 13 x = 39 X x 3 = x 3 =9 Calculation: Beyond Counting Multiplication: 13 x = 39 X Using the Place Value Counters, how would you represent the calculation above? Calculation: Beyond Counting Multiplication: 13 x =39 13 x X Teaching and practicing times tables Understanding the duality of multiplication! Relationship with division. Using a counting stick or line. Variety of practice methods Fluency/reasoning Recitation Rapid recall Calculation: Beyond Counting Systematic Variation: variety of tasks in a systematic manner while keeping the concept constant. 55 17 = 56 18 = 57 19 = 58 20 = What do you notice? Can you express this as a general rule? How could you solve this calculation now? 24 = 64 26 = 66 28 = 68 30 = 73 49 = 74 50 = 86 48 = 88 50 = Partitioning Near doubles Adjusting Scaling Known facts Estimating Calculation: Beyond Counting Reasoning based strategies Powerful questions to ask to support fluency (esp. in mental calculation!) 1.What do you notice? THINK before acting! of 48 Powerful questions to ask to support fluency (esp. in mental calculation!) 1.What do you notice? THINK before acting! 2.Whats the same and different? 3.Can you do it another way? 4.What if? 5.Can you find the mistake? 6.Does this answer look reasonable? 7.Is this true/false? How do you know? 8.What models would help you teach this to a younger child. 9.How could you prove it! Calculation Paper SATs No equipment x1 calculation paper 25 questions in approx 20 mins No formal timing. x1 reasoning and problem solving paper No equipment x1 calculation paper 36 questions in 30 mins No mental maths or calculator papers! x2 reasoning and problem solving papers How can you help? Positive attitude to the maths learning. Use everyday opportunities to practise maths skills. Praise the process and not just a correct answer. Find the logic in what they are saying! No wrong strategy (if it gets the right answer!) BUT some strategies more helpful than others! Use powerful questions when exploring maths problems. Praise deep thinking skills (Super Powers) rather than speed. Embrace the mistakes!