2 Finding rules and patterns - NIM A game for two playersStart with 20 countersEach player can remove 1,2,3, counters in turnThe loser is the person who picks up the last counter.Nim probably originated in China, the current name of this game is a loan word from German verb Nimm – to take).The theory of the game was discovered by a maths professor at havard Uni 1901.Nim is a simple game with finite possibilities, however there is a tremendous variety in the games implementation.Versions of Nim can be played with from one to at least a dozen row and the number of counters in a row can very from one to as many as two dozen. Some versions require winner takes last object, others involve winner avoids taking last object.My version (from Carol Vorderman’s book)Look for number pattern 1,5,9,13, 17 – if you leave your partner with this number of counters you are sure to win.
3 We are preparing you to teach mathematics by : Discussing the importance of subject knowledge and pedagogical knowledge in the teaching and learning of mathematicsConsidering the importance of early counting for all learnersConsidering how arithmetic can be taught through using and applying activities
4 New standards: Standard 3 Demonstrate good subject and curriculum knowledgehave a secure knowledge of the relevant subject(s) and curriculum areas, foster and maintain pupils’ interest in the subject, and address misunderstandingsdemonstrate a critical understanding of developments in the subject and curriculum areas, and promote the value of scholarshipif teaching early mathematics, demonstrate a clear understanding of appropriate teaching strategies.
5 The Aims of The New Curriculum The latest draft National Curriculum for mathematics aims to ensure 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 problemsreason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical languagecan 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.These three aims should not be lost in the detail of the programmes of study. The new draft seeks to strengthen these aims.These aims come from research carried out by the DfE into high performing jurisdictionshttps://www.education.gov.uk/publications/standard/publicationDetail/Page1/DFE-RR178
6 Conceptual Understanding BALANCEProcedural FluencyConceptual UnderstandingTalk about how there needs to be a balance of the two, but more so than this, the two need to be integrated.Many schools are doing either one or the other well but few are doing both, this is the challengeIs this the right image – discuss and see the next slide. We want to emphasise the integrationINTEGRATION6
7 FluencyThe government wishes to continue to emphasise fluency, but this should not be understood to mean “rote learning without understanding”.....conceptual understanding is clearly important and ..any emphasis on practice needs to be a part of achieving that understanding.Stefano Pozzi Mathematics in School May 2013 p2
8 Many secondary teachers PedagogyClassroom practiceMany secondary teachersMany primary teachersThe best teachersSubject knowledgeLisaOfSTED (2008) Understanding the Score
9 Make a list of things 1) Children need to know in order to calculate 32 –
10 Principles of Counting Gelman and Gallistel (1986) One to one principle – giving each item in a set a different counting word. Synchronising saying words and pointing.Stable order principle - Keeping track of objects counted knowing that numbers stay in the same order.Cardinal principle – recognising that the number associated with last object touched is the total number of object. The answer to ‘how many?’Abstraction principle - recognising small numbers without counting them and counting things you cannot move or touch.Order irrelevance principle - counting objects of different sizes and recognising that if a group of objects is rearranged then the number of them remains the same.
13 0-99 or 1-100 Midge Pasternack http://www. atm. org 234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495969798991001-100 rules OK Ian Thompson123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899Jigsaw: Cut up into pieces to make a jigsaw for the children to reassemble. Increase the number of pieces to make it harder.Inkblots Number gridTotal 100: Find pairs of numbers on the hundred square that total 100. How many different pairs can you find? How could you organise your answers so that you know you have found all of the possible ways? Consecutive numbers: Circle three numbers next to each other in a row. Find their total. Repeat for other groups of three consecutive numbers. What do all of the answers have in common? Try to explain why this happens.Predictions: Cover the multiples of 3 up to 30. Use the pattern to predict whether the number 52 will be in the sequence. Try predicting other numbers. How do you know? How could you check your answer? Repeat the activity using different multiplesSquares:Draw a 2 by 2 square on the hundred square.Add the numbers in opposite corners.What do you notice? Is it the same for different 2 by 2 squares?Now multiply the numbers in opposite corners. What do you notice this time? Is it always true?1-100 gridcardinal modelBetter representation of complements to 10 and 1001–100 square shows all two-digit numbers as comprising the number of rows denoted by the first digit plus the number of individual squares denoted by the second● Counting● 1–10 number strip● 1–100 square● 0–99 square● Number line● Empty number line.0-99Represents ordinal model‘0’ is given a prominent position, and children can learn early on about whole numbers (rather than counting numbers) and, place value, a concept very difficult for many, can be illustrated.rounding is easier to understandSubtraction eg
14 Common errors in counting in KS1 Counting one, two, three then any number name or other name to represent manyNumber names not remembered in orderCounting not co-ordinated with partitionCount does not stop appropriatelyCounts an item more than once or not at allDoes not recognise final number of count as how many objects there areCounting the start number when ‘counting on’ rather than the intervals (jumps) when ‘counting on’ on a number line.
15 Common errors in counting in KS2 when counting on or back, include the given number in their counting rather than starting from the next or previous number or counting the ‘jumps’;Difficulty counting from starting numbers other than zero and when counting backwards;understand the patterns of the digits within a decade, e.g. 30, 31, 32, ..., 39 but struggle to recall the next multiple of 10 (similarly for 100s);Know how to count on and count back but not understand which is more efficient for a given pair of numbers (e.g by counting on from 19 but 22-3 by counting back 3);Not understanding how place value applies to counting in decimals e.g. 0.8, 0.9, 0.10, 0.11 rather than 0.8, 0.9, 1.0, 1.1;Counting upwards in negative numbers as -1, -2, -3 … rather than -3, -2, -1…
17 Be Nasty Draw a grid big enough for digit cards Player Hundreds Tens Units (ones)Player 1Player 2Lisa
18 Be NastyLO: To use knowledge of place value to order numbers up to 1000RulesShuffle the number cards place face down in a stackTake turns to pick up a number card. You can place your number card on your own HTU line or on your partner’s HTU line.The aim is to make your own number as close as possible to the target – and to stop your partner making a number closer to the target.Take it in turns to go first.
19 Be Nasty Largest number Smallest number Nearest to 500 Nearest to a multiple of 10Nearest to a multiple of 5Nearest to a square numberNearest any centuryLowest even numberNearest odd number to 350
20 From Counting to Addition: 2 + 3 = 5 Count allCount on from firstCount on from largerCount on from eitherKnown factDerived facts1,2...1,2,3..1,2,3,4,53,4,54,5 3,4,5 or 4,5 52+3 =5 so 3+3 =6 and =2Carpenter and Moser (1983)
21 What’s the difference between…..? Tom had two sweets and John had threesweets how many did they have altogether?Tom had two sweets and bought three more. How many sweets does he have now?
22 Conceptual structures for addition Aggregation - combining of two or more quantities (How much/many altogether? What is the total?Tom had two sweetsand John had threesweets how manydid they have altogether?Augmentation – where one quantity is increased by some amount (increase by)Tom had two sweets and bought two more. How many sweets does he have now
23 Conceptual structures for subtraction Partition/change/take away - Where a quantity is partitioned off in some way and subtraction is required to calculate how many or how much remains. (Take away, How many left? How many are/do not?)Tom had five sweets,John ate three sweets. Howmany sweets did Tom have left?Comparison – a comparison is made between two quantities. (How any more?How many less/fewer? Howmuch greater? How muchsmaller? Tom had 5 sweets, Johnhad three sweets. How many moresweets did Tom have than John?
24 From Counting to Subtraction: 9 – 4 = 5 Ian Thompson (1997) Teaching and Learning Early Number, OUPCount out:Count back from:Count back to:Count up to:Recall of known number factsUse derived facts:Hold up 9 fingers (1, 2, 3, ...., 9) and fold down 4 (1, 2, 3, 4). Count the remaining 5 fingers (1, 2, 3, 4, 5).Count back 4 numbers from 9: ‘8, 7, 6, 5’Count back from first number to second saying: ‘8, 7, 6, 5, 4’ and tallying the numbers said (1, 2, 3, 4, 5)Count on from 4 up to 9: ‘5, 6, 7, 8, 9’to 10, 20 …, ways of making 100, 1000 e.g = 15 so = 14
25 Conceptual structures for multiplication Repeated addition - ‘so many sets of’ or ‘so many lots of’This is four lots of two this is written as 2 x 4Scaling structure – increasing a quantity by a scale factor (doubling, so many times bigger...so many times as much as). Tom has three times as many sweets as John.John Tom
26 Conceptual structures for division Equal sharing- (shared between, divided by) There are 8 sweets shared between four children. How many sweets do they get each?Equal grouping - I want to buy 8 sweets they come in packs of two . How many packs must I buy.
27 Calculation laws Commutative law - axb = bxa and a+b = b+a eg 3 x 4 = 4 x 3=Associative law - (axb) x c = a x (bxc)eg 24 x 6 = (4x6) x 6 = 4x (6x6)(5 + 7) + 3 = 5 + (7 + 3)Distributive law or partitioning (a+b) x ceg 12 x 7 = (10 x 7) + ( 2 x 7)and 84 ÷ 7 = (70 ÷ 7) + (14 ÷ 7)
30 Magic squaresUsing the digits 1- 9 arrange them in the 3 x 3 grid so that each row, column and diagonal adds up to the same amount.What would happen if you added two to each number - would the square still be magic? What could you tell you partner about the magic square now.What learning was going on? In pairs or small groups, consider the magic square activity in relation to Skemp’s theory of relational and instrumental understanding.30
31 Bruner (1961) Discovery Learning Based on constructivist learning theory and problem solving.Learner actively constructs knowledge and skills rather than passively receiving knowledge from a teacher/text book or equivalent.Learning is more effective when a student is actively engaged in the learning processPupils retain knowledge and have deeper understanding if they discover it for themselveslearning builds upon prior knowledge and understandingPedagogical aims:Promote "deep" learningPromote meta-cognitive skills (develop problem-solving skills, creativity, independent learning , evaluation)Promote student engagement.
32 Developing higher order thinking Bloom's Taxonomy is a hierarchy of skills that reflects growing complexity and ability to use higher-order thinking skills (HOTS).Bloom, B.S. (Ed.) (1956) Taxonomy of educational objectives: The classification of educational goals: Handbook I, cognitive domain. New York ; Toronto: Longmans, Green.Higher order thinking skills are not just abot mathematical content knowledge. Just as it is possible to engage in very hard questions that involve a high level of content knowledge but very problem solving skills it is also possible to engage in activities that only need a low level of mathematical content knowledge. In the former you are going to need well tuned knowledge level skills and in the latter HOTS
33 How can we encourage higher order thinking skills? If children spend most of their time practising paper and pencil skills on worksheet exercises, they are likely to become faster at executing these skills.If they spend most of their time watching the teacher demonstrate methods for solving special kinds of problems, they are likely to become better at imitating these methods on similar problems.If they spend most of their time reflecting on how various ideas and procedures are the same or different, on how what they already know relates to the situations they encounter, they are likely to build new relationships. That is, they are likely to construct new understandings.Skemp instrumental and relational understanding (1976)Ryan & Williams surface and deep understanding (2007)Hiebert ( 1993)
34 Designing mathematical tasks Mathematical reasoning, even more so than children’s knowledge of arithmetic, is important for children’s later achievement in mathematics (Nunes et al 2009 p.3)Nunes et al (2009) define reasoning as ‘learning to reason about the underlying relations mathematical problems they have to solve’
35 But surely only the most able children can reason……. “If teachers consider that tasks involving mathematics thinking are suitable for ‘high attainers’ then the result may be that ‘low attainers’ are given a diet of routine and repetitive tasks on which they have already demonstrated their low attainment. But if all learners are treated as possessing the powers necessary to think mathematically, and if those powers are evoked, developed and refined, the so called ‘low attainers’ can transcend expectations (Mason and Johnston-Wilder ( )
36 Sudoko As a group on the floor with laminated shape cards Or in pairs with small cardsLisa, I have a bag of resources which I will bring so at this point we can share ideas. Can you bring your one is a snail book please.
39 x3240492215272442Fit into the purple boxes: 2,3,4,5,6,7,8,9,10,11,12One number has to be used twice! Which one? Why?
40 Always Sometimes never Shepway Teaching Schools Alliance, Unit 31a Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH
41 EYFSHow many different patterns of dots can you make with five dots?Year 1When you add two numbers, you can change the order of the numbers and the answer will be the sameYou can make 4 different two digit numbers with the digits 2 and 3: 23, 32, 22, 33When you add 10 to a number the units digit stays the same.Year 2When you subtract ten from a number, the units digit stays the sameYou can add 9 to a number by adding 10 and subtracting 1All even numbers end in 0, 2, 4, 6, 8If you have 3 digits, and use each one exactly once in a three digit number, you can make 6 different three digit numbers
42 Questioning‘The subtle art of questioning is the art of teaching. In a real sense, learning to teach is learning to ask questions.’Tanner and Jones (2000)
43 Improving your questioning Examples of open-ended questions that invite children to think includeWhat do you think…………?How do you know………….?Why do you think that………..?Do you have a reason………..?Is this always so……….?Is there another way/reason/idea…………..?What if………….? What if…….does not…..?Where is there another example of this…..?What do you think happens next?
44 Instead ofFind the perimeter of a 3x8 rectangleYou could askIf the area of a rectangle is 24cm2 what is the perimeter?Improve these QuestionsA chew costs 3p and a lolly costs 7p. What do they cost together?What is 6- 4?Is 16 an even number?What are 4 threes?What is this shape called?
45 Effective questioning strategies to promote thinking Sequencing a set of questionsPitching appropriatelyDistributing questions around the classPrompting and probingListening and responding positively – inviting further questionsChallenging right as well as wrong and underdeveloped answersUsing written questions effectively.