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 pupil’s 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 isn’t enough if students don’t 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 Tθ 1 52

Calculation: Beyond Counting Have a go counting with “Base 5” ! Try writing 1 – 20! Tw Fθ

Calculation: Beyond Counting What number is this? Tw Fθ 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: … don’t 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.What’s 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 SAT’s

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!