Shepway Teaching Schools Alliance
31a Folkestone Enterprise Centre,
Shearway Road, Folkestone CT19 4RH
Telephone Number: 01303 298266
e-mail: office@shepwayts.co.uk,
website: www.shepwayts.co.uk
Finding rules and patterns - NIM
*A game for two players
*Start with 20 counters
*Each player can
remove 1,2,3, counters
in turn
*The loser is the person
who picks up the last
counter.
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
Considering how arithmetic can be taught
through using and applying activities
New standards: Standard 3
Demonstrate 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 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.
Procedural
Fluency
Conceptual
Understanding
INTEGRATION
*
* The 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
Many secondary
teachers
Classroom
practice
Subject
knowledge
The best teachers
Pedagogy
Many primary
teachers
OfSTED (2008) Understanding the Score
http://www.ofsted.gov.uk/resources/mathematics-understanding-score
*
32 – 3
32 - 29
1.
One to one principle – giving each item in a set a different
counting word. Synchronising saying words and pointing.
2.
Stable order principle - Keeping track of objects counted
knowing that numbers stay in the same order.
3.
Cardinal principle – recognising that the number associated with
last object touched is the total number of object. The answer to
‘how many?’
4.
Abstraction principle - recognising small numbers without
counting them and counting things you cannot move or touch.
5.
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.
http://webarchive.nationalarchives.gov.uk/20110202093118/http:/national
strategies.standards.dcsf.gov.uk/node/84889
0
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0-99 or 1-100 Midge Pasternack
http://www.atm.org.uk/journal/archi
ve/mt182files/ATM-MT182-3435.pdf
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1-100 rules OK Ian Thompson
http://www.atm.org.uk/journal/archi
ve/mt185files/ATM-MT185-1415.pdf
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http://www.teachfind.com/nati
onal-strategies/mathematicsitp-number-grid
*
 Counting one, two, three then any number name or other
name to represent many
 Number names not remembered in order
 Counting not co-ordinated with partition
 Count does not stop appropriately
 Counts an item more than once or not at all
 Does not recognise final number of count as how many
objects there are
 Counting 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
Player
Player 1
Player 2
Hundreds
Tens
Units (ones)
*
LO: To use knowledge of place value to order
numbers up to 1000
Rules
• Shuffle the number cards place face down in a
stack
• Take 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.
*
* Largest number
* Smallest number
* Nearest to 500
* Nearest to a multiple of 10
* Nearest to a multiple of 5
* Nearest to a square number
* Nearest any century
* Lowest even number
* Nearest odd number to 350
*
1,2...1,2,3..1,2,3,4,5
3,4,5
4,5
3,4,5 or 4,5
5
2+3 =5 so 3+3 =6 and
5-3=2
Carpenter and Moser (1983)
*
* 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?
*
Ian Thompson (1997) Teaching and Learning Early Number, OUP
*Count out:
*Count back from:
*Count back to:
*Count up to:
*Recall of known
number facts
*Use 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. 20 - 5 = 15 so 20
- 6 = 14
*
• Repeated addition
- ‘so many sets of’ or ‘so many lots of’
This is four lots of two this is written as 2 x 4
• Scaling 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
*
• 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.
*
*Commutative law
- axb = bxa and a+b = b+a
eg 3 x 4 = 4 x 3
55 + 45 = 45 + 55
*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 c
eg 12 x 7 = (10 x 7) + ( 2 x 7)
and 84 ÷ 7 = (70 ÷ 7) + (14 ÷ 7)
http://dera.ioe.ac.uk/778/
*
*
* 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.
* 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.
*
• 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 process
Pupils retain knowledge and have deeper understanding if they
discover it for themselves
learning builds upon prior knowledge and understanding
Pedagogical aims:
Promote "deep" learning
Promote meta-cognitive skills (develop problem-solving skills, creativity,
independent
learning ,
evaluation)
Promote student engagement.
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.
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.
Hiebert ( 1993)
*
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)
*
“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
(2006 . 41)
*
1
2
3
4
*Fit into the purple
x
32
boxes:
2,3,4,5,6,7,8,9,10,11,12
40
*One number has to be
49
22
15
27
24
42
used twice!
Which one?
Why?
*
Shepway Teaching Schools Alliance, Unit 31a
Folkestone Enterprise Centre, Shearway Road,
Folkestone CT19 4RH
*EYFS
*How many different patterns of dots can you make
with five dots?
*Year 1
*When you add two numbers, you can change the order
of the numbers and the answer will be the same
*You can make 4 different two digit numbers with the
digits 2 and 3: 23, 32, 22, 33
*When you add 10 to a number the units digit stays the
same.
*Year 2
*When you subtract ten from a number, the units digit
stays the same
*You can add 9 to a number by adding 10 and
subtracting 1
*All even numbers end in 0, 2, 4, 6, 8
*If you have 3 digits, and use each one exactly once in
a three digit number, you can make 6 different three
digit numbers
*
‘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)
*
Examples of open-ended questions that invite children to
think include
*What 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?
*
*
Instead of
* Find the perimeter of a 3x8 rectangle
*
You could ask
* If the area of a rectangle is 24cm2 what is the perimeter?
Improve these Questions
1. A chew costs 3p and a lolly costs 7p. What do they cost
together?
2. What is 6- 4?
3. Is 16 an even number?
4. What are 4 threes?
5. What is this shape called?
*
*Sequencing a set of questions
*Pitching appropriately
*Distributing questions around the class
*Prompting and probing
*Listening and responding positively – inviting
further questions
*Challenging right as well as wrong and
underdeveloped answers
*Using written questions effectively.