Numeral ID Year 2...Image source: haiku deck Michelle Tregoning, 2016 Mapping backwards to make connections Michelle Tregoning, 2016 Michelle Tregoning, 2016 Making connections: Relating

Becoming

mathematicians:

Making meaningful connections

Michelle Tregoning

28 552

Michelle Tregoning, 2016

Michelle Tregoning, 2016 Image source: haiku deck


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?




Image source: K/1H Laguna Street Public School S. Hughes Michelle Tregoning, 2016


28 552



Image source: haiku deck Michelle Tregoning, 2016


1457

597 481


In Kindergarten:

• 86% of students are on track across NSW

• 14% are not on track…that’s 1387 young people

In Year 1:


• 15% are not on track…that’s 1411 young people

In Year 2:


• 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?



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?


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?


* 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


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?


‘The foundations for some later concepts are being laid years

before full understanding of the concept may manifest itself.’ Hurst and Hurrell, 2013


Mapping

backwards to make

connections



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



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



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



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


Image source: haiku deck Jo Boaler


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


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




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




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


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



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 Image source: haiku deck Walt Whitman


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




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)




Year 2

Green:

• Place value 1 or higher

Amber:

• Place value 0 / EAS COB or higher

Red:

• EAS Lower than counting-on-and-back



Becoming

mathematicians:

Making meaningful connections



Image source: haiku deck Quote: Putting faces on the data



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)



Adapted by M. Tregoning from: MeE Framework, Munns and Sawyer

Pedagogy and engagement



.

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,)


"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


Documents

Numeral ID Year 2...Image source: haiku deck Michelle Tregoning, 2016 Mapping backwards to make connections Michelle Tregoning, 2016 Michelle Tregoning, 2016 Making connections: Relating