numeral id year 2...image source: haiku deck michelle tregoning, 2016 mapping backwards to make...
TRANSCRIPT
Becoming
mathematicians:
Making meaningful connections
Michelle Tregoning
28 552
Michelle Tregoning, 2016
Michelle Tregoning, 2016 Image source: haiku deck
Michelle Tregoning, 2016
A teacher needed to organise
transport for an enormous
festival taking place across
NSW. In total, 28 552 students
were involved.
Travelling on 56 seat buses,
how many coaches were
needed?
Michelle Tregoning, 2016 Image source: haiku deck
Michelle Tregoning, 2016 Image source: haiku deck
Michelle Tregoning, 2016 Image source: haiku deck
Image source: K/1H Laguna Street Public School S. Hughes Michelle Tregoning, 2016
Michelle Tregoning, 2016 Image source: haiku deck
28 552
Michelle Tregoning, 2016
Michelle Tregoning, 2016
Image source: haiku deck Michelle Tregoning, 2016
Michelle Tregoning, 2016
1457
597 481
Michelle Tregoning, 2016
In Kindergarten:
• 86% of students are on track across NSW
• 14% are not on track…that’s 1387 young people
In Year 1:
• 85% of students are on track across NSW
• 15% are not on track…that’s 1411 young people
In Year 2:
• 67% of students are on track across NSW
• 33% are not on track…that’s 3058 young people who have 6 weeks of schooling left in 2016
• Did you notice the same trend in reading, comprehension and writing?
Image source: haiku deck Michelle Tregoning, 2016
What might some
of the ‘why’s’ be?
Michelle Tregoning, 2016
Michelle Tregoning, 2016 Image source: haiku deck
How would you work out
32 – 19? How many different ways can you think of to work out a
solution?
How would could you represent those strategies so your
thinking makes sense to someone else?
Michelle Tregoning, 2016
How would you work out
32 – 19? How many different ways can you think of to work out a
solution?
How would could you represent those strategies so your
thinking makes sense to someone else?
Michelle Tregoning, 2016
* uses a range of strategies and informal recording
methods for addition and subtraction involving one-
and two-digit numbers MA1-5NA, MA1-1WM, MA1-2WM, MA1-3WM
Michelle Tregoning, 2016
32 - 19
What did I need to understand and know in order to solve the
problem?
• Number sense
• I needed to think about the numbers:
• What is the relationship between them?
• What numbers are easy for me to work with?
• ‘Landmark’ numbers
• What facts do I know that I can use?
• I control the numbers – I can take advantage of what I
know to reason my way to a solution
• Operational ‘sense’
• I needed to think about the operations:
• What mathematical properties can I apply in this context?
• ‘Equivalence is about balance’
Where can this
take me?
Michelle Tregoning, 2016
‘The foundations for some later concepts are being laid years
before full understanding of the concept may manifest itself.’ Hurst and Hurrell, 2013
Michelle Tregoning, 2016 Image source: haiku deck
Mapping
backwards to make
connections
Michelle Tregoning, 2016
Michelle Tregoning, 2016
Making connections: Relating and
using relationships
Every number is unique…whilst also being related to other
numbers in many ways.
• Half of 24
• Double 6
• 2 more than 10
• 8 less than 20
• A quarter of 12 is 3
• A third of 12 is 4
• 3 fours is the same as 12
• 4 threes is the same as 12
• A tenth of 120
• 10 times larger than 1.2
12 Counting with understanding
involves counting with one-to-
one correspondence,
recognising that the last
number name represents the
total number in the collection,
and developing a sense of the
size of numbers, their order
and their relationships.
Representing numbers in a
variety of ways is essential for
developing number sense. Background information: Mae-4NA
Michelle Tregoning, 2016
Making connections: Relating and
using relationships
Estimating and benchmarks: benchmarks are useful in all
mathematics. Estimating helps us develop a tolerance for error
and uncertainty and forms a significant part of critical numeracy.
estimate the number of objects in a group of up to 20 objects, and count to check MAe-4NA
Image source: Great Estimations
Michelle Tregoning, 2016
Making connections: Relating and
using relationships
Estimating and benchmarks: benchmarks are useful in all
mathematics. Estimating helps us develop a tolerance for error
and uncertainty and forms a significant part of critical numeracy.
Image source: Number SENSE
Michelle Tregoning, 2016
Making connections: Relating and
using relationships
Every number (and composite unit) is flexible – it can be
partitioned in many different ways.
• Which of your ways of partitioning 19 is most useful when…
• you want to combine it with 62?
• you want to subtract it from 27?
• you want to work out the difference between 19 and 43?
• Which of your partitions of 19 is least useful when…
• Which of your partitions of 19 would you change if you wanted
to…
19 model and record patterns for
individual numbers by making all
possible whole-number
combinations MA1-8NA
Michelle Tregoning, 2016
Image source: haiku deck Jo Boaler
Michelle Tregoning, 2016
Making connections: Patterns
underpin everything
A core foundation of being mathematical is the ability to
generalise. Stemming from the ability to identify patterns,
generalising forms a vital foundation of later algebra.
Wonderment and exploration are at the heart of generalising.
• There must be an element of repetition
• Can be represented in different ways
• Many kinds of patterns – repeating, growing, shrinking, etc.
patterns
‘same-ness’ equivalence
attributes
flexibility
sort and classify
describe
extend
translate (match)
notice test
Recognises, describes and
continues repeating patterns
MAe-8NA, MAe-1WM, MAe-
2WM, MAe-3WM
Michelle Tregoning, 2016
Making connections: REALLY
understanding the operations
What do you already know? How are the operations related?
What connects them? What distinguishes between them? How will
you design learning where students explore and investigate these
properties (rather than being ‘told’)?
• You can add and multiply numbers in any order you like (the commutative property) but when dividing and subtracting, the order matters
• This is because in some cases you are creating a new total and in others, you start from the total
• Addition and subtraction are inverse operations. So are multiplication and division
• You can +, -, x and ÷ in parts (you can partition numbers and composite units – in multiplicative situations, we call this the distributive property)
• Numbers can be adjusted to suit the ‘mathematician’:
• + / x: 16 + 35 = 11 + 40; 16 x 35 = 8 x 70 = 2 x 280
• -/ ÷: 35 – 16 = 35 – 16 = 39 – 20; 30 ÷ 6 = 15 ÷ 3
Michelle Tregoning, 2016
Making connections: REALLY
understanding the operations
When you add 1, the sum is the next number in the counting
sequence. This works for composite units (like 10s) and in reverse
for subtraction.
one-to-one
stable-order
cardinal
order-irrelevance
abstraction
Counting with understanding involves counting with one-to-one correspondence, recognising that the last number name represents the total number in the collection, and developing a sense of the size of numbers, their order and their relationships. Representing numbers in a variety of ways is essential for developing number sense. Background information: Mae-4NA
Michelle Tregoning, 2016
Making connections: REALLY
understanding the operations
There are different addition and subtraction situations. Some
strategies work in some contexts and not in others. Some situations
make it easier to work out in my head than others…and it’s based
on what I know and what I understand.
a. 17 + ___ = 34 b. 80 + 30 c. 7 + 15 + 4
d. 10 + 10 + 10 + 10 e. 25 + 25 f. 38 + ___ = 66
g. 65 - ___ = 20 h. 14 – 9 i. 18 – 2 – 2
Which of these would I work out in my head? Which would I use
concrete materials or drawings to help work out a solution?
Adapted from: Number SENSE
28 552
Michelle Tregoning, 2016
Maths is ‘a subject that's all about different
connections between ideas. And it's those connections that mathematicians and
others regard as beautiful. So if we want students to understand maths deeply and to appreciate that beauty in maths, that
which is a great basis for developing all of their mathematical thinking, then we really
need to encourage a connected
approach to the subject.’
Jo Boaler
Michelle Tregoning, 2016
Michelle Tregoning, 2016
A brief intermission for some key ideas
• The foundations of later maths are built from Kindergarten
• Understanding patterns and relationships (between numbers and operations)
are critical
• Laying good foundations takes time, persistence and creativity to continue
making it focused, meaningful and engaging
• Mathematics is about meaning making: we need to support our students and
our colleagues in making meaningful connections
• Communication is critical – visual, concrete, verbal, symbolic
• Something appears to be going on by the time our students reach Year 2
• Do they deeply understand counting sequences (FW and BW, 1s and 10s)?
• Do they have strong number sense?
• Do they really know and understand just how flexible mathematics is?
• Do they understand the operations?
• Do they think about what they know and the context when solving problems?
• Do they have meaningful and frequent opportunities to notice, wonder, generalise, test, communicate, debate, visualise, model and compare?
• Do they understand patterns?
• Do we need to better support….?
School analysis
(with the guidance of Peter Gould)
Michelle Tregoning, 2016
Michelle Tregoning, 2016 Image source: haiku deck Walt Whitman
Michelle Tregoning, 2016
Kindergarten
Green:
• Early Arithmetical Strategies (EAS) perceptual or higher
• Forward Number Word Sequences (FNWS) Facile (30) or
higher
Amber:
• EAS Perceptual
• FNWS Initial (10), Intermediate (10) or Facile (10) (i.e. L1, L2 or
L3)
Red:
• EAS Emergent
Michelle Tregoning, 2016
Michelle Tregoning, 2016
Michelle Tregoning, 2016
Year 1
Green:
• Early Arithmetical Strategies (EAS) figurative or higher
Amber:
• EAS Perceptual
• FNWS Facile (30) or higher
Red:
• The rest (Rationale: EAS Perceptual + FNWS Facile (30) is the
expectation of end of Kindergarten)
Michelle Tregoning, 2016
Michelle Tregoning, 2016
Michelle Tregoning, 2016
Year 2
Green:
• Place value 1 or higher
Amber:
• Place value 0 / EAS COB or higher
Red:
• EAS Lower than counting-on-and-back
Michelle Tregoning, 2016
Michelle Tregoning, 2016
Becoming
mathematicians:
Making meaningful connections
Michelle Tregoning, 2016 Image source: haiku deck
Michelle Tregoning, 2016
Image source: haiku deck Quote: Putting faces on the data
Michelle Tregoning, 2016 Image source: haiku deck
Michelle Tregoning, 2016
The connection to emotion
Individuals’ attitudes, beliefs and emotions play a significant role in their interest and response to mathematics in general, and their employment of mathematics in their individual lives.
Students who feel more confident with mathematics, for example, are more likely than others to use mathematics in the
various contexts that they encounter. Students who have positive emotions towards mathematics are in a position to learn mathematics better than students who feel anxiety towards that subject. Therefore, one goal of mathematics education is for students to develop attitudes, beliefs and
emotions that make them more likely to successfully use the mathematics they know, and to learn more mathematics, for
personal and social benefit. (OECD, 2013)
Michelle Tregoning, 2016 Image source: haiku deck
Michelle Tregoning, 2016
Adapted by M. Tregoning from: MeE Framework, Munns and Sawyer
Pedagogy and engagement
Michelle Tregoning, 2016 Image source: haiku deck
Michelle Tregoning, 2016 Image source: haiku deck
.
Becoming a proficient mathematician requires working with all of the mathematical
proficiencies – fluency, problem solving, reasoning and understanding – from the
beginning. And by mathematician here I mean anyone using mathematics in his or her life.
Everyone is a mathematician.
(Askew, 2012,)
Michelle Tregoning, 2016
"The principle goal of education in the schools should be creating men and
women who are capable of doing new things, not simply repeating what
other generations have done; men and women who are creative,
inventive and discoverers, who can be critical and verify, and not accept,
everything they are offered.” Piaget
Michelle Tregoning, 2016