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Preparing to teach mathematics. We are preparing you to teach mathematics by : Discussing the importance of subject knowledge and pedagogical knowledge in the teaching and learning of mathematics Considering the importance of early counting for all learners - PowerPoint PPT Presentation

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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 the aims of the National Curriculum

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.

Using 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.

Arithmetic fluency:Learning mathematical procedures and skills and using this knowledge to solve problemsReasoning: learning to reason about the underlying relations in mathematical problems they have to solveLearning and remembering skills and procedures

Calculating efficiently

Remembering mathematical vocabulary

Remembering facts

EgKnowing how to addArguing, communicating

Problem solving

Investigating

Thinking mathematically

Understanding ideas or concepts

EgKnowing when to add

To 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 To reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language To 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. https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/184064/DFE-RR178.pdf

Procedural FluencyConceptual UnderstandingINTEGRATION

32 3 32 - 29

Subject knowledgeOfSTED (2008) Understanding the Score http://www.ofsted.gov.uk/resources/mathematics-understanding-score

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.

Ordering numbersMore than, less thanCounting out a given numberCounting from a given numberReciting number names in order and becoming consistent, including through decade and hundred changes Reciting number names with decimals and fractionsOrdering numbers including with fractions and decimals

Singapore MathsConcrete Pictorial Abstract

Bruners phases of learningEnactiveIconicSymbolic

5

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.

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. 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

Draw a grid big enough for digit cards

PlayerHundreds Tens Units (ones)Player 1Player 2

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.

Largest 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 350

Positional- the quantities represented by the individual digits are determined by the positions that they hold in the whole numeral. The value given to a digit is according to the position in a numberBase 10: the value of the position increases in powers of 10Multiplicative; the value of an individual digit is found by multiplying the face value of the digit by the value assigned to its position.Additive: the quantity represented by the whole numeral is the sum of the values represented by the individual digits (Ross 1989)

Twenty eight, twenty nine, twenty tenWriting 10016 for 116Writing 1.6 for 1.06Placing 1.35 as larger on a number line than 1.5Lining numbers up incorrectly in column additionWriting the sequence 1.7, 1.8, 1.9, 1.10, 1.11..

1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900

1000 2000 3000 4000 5000 6000 7000 8000 9000

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?

Aggregation - combining of two or more quantities (How much/many altogether? What is the total?Tom had two sweets and John had three sweets how many did 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

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. How many sweets did Tom have left?

Comparison a comparison is made between two quantities. (How any more? How many less/fewer? How much greater? How much smaller? Tom had 5 sweets, John had three sweets. How many more sweets did Tom have than John?

Counting forwards and backwardsOne more than, one less thanCounting on or back in steps of 2,5 and 10Counting on or back from the larger numberPartitioning numbers into 5 and a bit e.g. 5 + 7 = 5 + 5 + 2Bridging through 10, using known facts to 10 e.g. 6 + 9 = (6 + 4) + 5; 15 9 = (15 5) - 4

Bridging through multiples of 10 e.g. 25 + 7 = (25 + 5) + 2; 22 5 = (22 2) 3Reordering numbers in addition e.g. 6 + 2 + 4 = 6 + 4 + 2Find differences by counting up e.g. 10 6 by counting 7, 8, 9, 10Using inverse operations e.g. 13 + 7 = 20 so 20 7 = 13Special cases: Using doubles facts to derive near doubles facts e.g. 6 + 6 = 12 so 6 + 8 = 14 and 6 + 5 = 11

Calculate 25 + 47

Using Dienes Using NumiconUsing Place Value Counters

Calculate 72 - 47

NumiconUsing Dienes Using Place Value counters

26 + 5725 + 2465 + 2973 - 6882 - 26156 99Then compare strategies with a friend.

0.31.67.24.60.210.55.72.365.38.30.125.27.32.710.741.99.23.92.39.86.22.636.1101.7

Different children prefer different mental calculation strategiesChoice of strategy may vary for different pairs of numbersThe choice of mental strategy for a particular pair of numbers is influenced by a range of factors:size of the numbers, personal preferences, size of the difference between the numbers, proximity of numbers to 10s or 100s numbers, special cases etc.

MentalWe may break the calculation into manageable parts eg 248 100 + 1 instead of 248 99We say the calculation to ourselves and so are aware of the numbers themselves eg 2000 10 is not much less than 2000WrittenWe never change the calculation to an equivalent one, 248 99 is done as it is

We dont say the numbers to ourselves, but talk about the digits instead saying 8 9 and 4 - 9

MentalWe usually begin with the most significant digitWe choose a strategy to fit the numbers eg 148 99 is not calculated in the same way as 84 77 although they are both subtractionsWe draw upon mathematical knowledge such as properties of numbers or number sense, learned facts etcWrittenWe usually begin with the least significant digitWe always use the same method

We draw upon the memory of a procedure although we may not understand how it works

Repeated addition - so many sets of or so many lots of

This is four lots of two this is writ