Preparing to teach mathematics

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Preparing to teach mathematics. Shepway Teaching Schools Alliance 31a Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH Telephone Number: 01303 298266 e-mail:, website: Finding rules and patterns - NIM. A game for two players - PowerPoint PPT Presentation

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Shepway Teaching Schools Alliance 31a Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RHTelephone Number: 01303 298266 e-mail:, website: to teach mathematics

10/16/2013 12:13 PM31b Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH. E-Mail: Website: game for two players

Start with 20 countersEach player can remove 1,2,3, counters in turnThe loser is the person who picks up the last counter.

Finding rules and patterns - NIMNim 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 Vordermans 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.

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

New standards: Standard 3Demonstrate good subject and curriculum knowledge have a secure knowledge of the relevant subject(s) and curriculum areas, foster and maintain pupils interest in the subject, and address misunderstandings demonstrate a critical understanding of developments in the subject and curriculum areas, and promote the value of scholarship if teaching early mathematics, demonstrate a clear understanding of appropriate teaching strategies.

The Aims of The New CurriculumThe 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 problems 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.

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 jurisdictions


Procedural FluencyConceptual UnderstandingINTEGRATIONTalk 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 integration6FluencyThe 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 p27Many secondary teachersMany primary teachersThe best teachersSubject knowledgeOfSTED (2008) Understanding the Score

Lisa Make a list of things

1) Children need to know in order to calculate

32 3 32 - 29

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.


0102030405060708090100Lisa12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485868788899091929394959697989910001234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495969798990-99 or 1-100 Midge Pasternack

1-100 rules OK Ian Thompson

Jigsaw: Cut up into pieces to make a jigsaw for the children to reassemble. Increase the number of pieces to make it harder. Inkblots Number grid 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 multiples Squares: 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 100 1100 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 110 number strip 1100 square 099 square Number line Empty number line.

0-99Represents ordinal model0 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 16 - 16

Common errors in counting in KS1Counting 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.

Common errors in counting in KS2when 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. 22-19 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

Be NastyDraw a grid big enough for digit cardsPlayerHundreds Tens Units (ones)Player 1Player 2

LisaBe NastyLO: To use knowledge of place value to order numbers up to 1000

RulesShuffle 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 partners 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.

Be NastyLargest numberSmallest number Nearest to 500Nearest to a multiple of 10Nearest to a multiple of 5Nearest to a square numberNearest any centuryLowest even numberNearest odd number to 350From Counting to Addition: 2 + 3 = 5 Count allCount on from firstCount on from largerCount on from eitherKnown factDerived facts

1,2...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 5-3=2

Carpenter and Moser (1983)Whats the difference between..?Tom had two sweets and John had three sweets how many did they have altogether?

Tom had two sweets and bought three more. How many sweets does he have now?

Conceptual structures for additionAggregation - combining of two or more